Generic graphs (common to directed/undirected)

This module implements the base class for graphs and digraphs, and methods that can be applied on both. Here is what it can do:

Basic Graph operations:

networkx_graph()	Create a new NetworkX graph from the Sage graph
igraph_graph()	Create a new igraph graph from the Sage graph
to_dictionary()	Create a dictionary encoding the graph.
copy()	Return a copy of the graph.
export_to_file()	Export the graph to a file.
adjacency_matrix()	Return the adjacency matrix of the (di)graph.
incidence_matrix()	Return an incidence matrix of the (di)graph
distance_matrix()	Return the distance matrix of the (strongly) connected (di)graph
weighted_adjacency_matrix()	Return the weighted adjacency matrix of the graph
kirchhoff_matrix()	Return the Kirchhoff matrix (a.k.a. the Laplacian) of the graph.
has_loops()	Return whether there are loops in the (di)graph
allows_loops()	Return whether loops are permitted in the (di)graph
allow_loops()	Change whether loops are permitted in the (di)graph
loops()	Return a list of all loops in the (di)graph
loop_edges()	Return a list of all loops in the (di)graph
number_of_loops()	Return the number of edges that are loops
loop_vertices()	Return a list of vertices with loops
remove_loops()	Remove loops on vertices in vertices. If vertices is None, removes all loops.
has_multiple_edges()	Return whether there are multiple edges in the (di)graph.
allows_multiple_edges()	Return whether multiple edges are permitted in the (di)graph.
allow_multiple_edges()	Change whether multiple edges are permitted in the (di)graph.
multiple_edges()	Return any multiple edges in the (di)graph.
name()	Return or sets the graph’s name.
is_immutable()	Return whether the graph is immutable.
weighted()	Whether the (di)graph is to be considered as a weighted (di)graph.
antisymmetric()	Test whether the graph is antisymmetric
density()	Return the density
order()	Return the number of vertices.
size()	Return the number of edges.
add_vertex()	Create an isolated vertex.
add_vertices()	Add vertices to the (di)graph from an iterable container
delete_vertex()	Delete a vertex, removing all incident edges.
delete_vertices()	Remove vertices from the (di)graph taken from an iterable container of vertices.
has_vertex()	Return True if vertex is one of the vertices of this graph.
random_vertex()	Return a random vertex of self.
random_vertex_iterator()	Return an iterator over random vertices of self.
random_edge()	Return a random edge of self.
random_edge_iterator()	Return an iterator over random edges of self.
vertex_boundary()	Return a list of all vertices in the external boundary of vertices1, intersected with vertices2.
set_vertices()	Associate arbitrary objects with each vertex
set_vertex()	Associate an arbitrary object with a vertex.
get_vertex()	Retrieve the object associated with a given vertex.
get_vertices()	Return a dictionary of the objects associated to each vertex.
vertex_iterator()	Return an iterator over the vertices.
neighbor_iterator()	Return an iterator over neighbors of vertex.
vertices()	Return a list of the vertices.
neighbors()	Return a list of neighbors (in and out if directed) of vertex.
merge_vertices()	Merge vertices.
add_edge()	Add an edge from u and v.
add_edges()	Add edges from an iterable container.
subdivide_edge()	Subdivide an edge \(k\) times.
subdivide_edges()	Subdivide k times edges from an iterable container.
delete_edge()	Delete the edge from u to v
delete_edges()	Delete edges from an iterable container.
delete_multiedge()	Delete all edges from u and v.
set_edge_label()	Set the edge label of a given edge.
has_edge()	Return True if (u, v) is an edge, False otherwise.
edges()	Return a list of edges.
edge_boundary()	Return a list of edges \((u,v,l)\) with \(u\) in vertices1
edge_iterator()	Return an iterator over edges.
edges_incident()	Return incident edges to some vertices.
edge_label()	Return the label of an edge.
edge_labels()	Return a list of edge labels.
remove_multiple_edges()	Remove all multiple edges, retaining one edge for each.
clear()	Empty the graph of vertices and edges and removes name, associated objects, and position information.
degree()	Return the degree (in + out for digraphs) of a vertex or of vertices.
average_degree()	Return the average degree of the graph.
degree_histogram()	Return a list, whose ith entry is the frequency of degree i.
degree_iterator()	Return an iterator over the degrees of the (di)graph.
degree_sequence()	Return the degree sequence of this (di)graph.
random_subgraph()	Return a random subgraph that contains each vertex with prob. p.
add_clique()	Add a clique to the graph with the given vertices.
add_cycle()	Add a cycle to the graph with the given vertices.
add_path()	Add a cycle to the graph with the given vertices.
complement()	Return the complement of the (di)graph.
line_graph()	Return the line graph of the (di)graph.
to_simple()	Return a simple version of itself (i.e., undirected and loops and multiple edges are removed).
disjoint_union()	Return the disjoint union of self and other.
union()	Return the union of self and other.
relabel()	Relabel the vertices of self
degree_to_cell()	Return the number of edges from vertex to an edge in cell.
subgraph()	Return the subgraph containing the given vertices and edges.
is_subgraph()	Test whether self is a subgraph of other.

Graph products:

cartesian_product()	Return the Cartesian product of self and other.
tensor_product()	Return the tensor product, also called the categorical product, of self and other.
lexicographic_product()	Return the lexicographic product of self and other.
strong_product()	Return the strong product of self and other.
disjunctive_product()	Return the disjunctive product of self and other.

Paths and cycles:

eulerian_orientation()	Return a DiGraph which is an Eulerian orientation of the current graph.
eulerian_circuit()	Return a list of edges forming an Eulerian circuit if one exists.
cycle_basis()	Return a list of cycles which form a basis of the cycle space of self.
all_paths()	Return a list of all paths (also lists) between a pair of vertices in the (di)graph.
triangles_count()	Return the number of triangles in the (di)graph.

Linear algebra:

spectrum()	Return a list of the eigenvalues of the adjacency matrix.
eigenvectors()	Return the right eigenvectors of the adjacency matrix of the graph.
eigenspaces()	Return the right eigenspaces of the adjacency matrix of the graph.

Some metrics:

cluster_triangles()	Return the number of triangles for nbunch of vertices as a dictionary keyed by vertex.
clustering_average()	Return the average clustering coefficient.
clustering_coeff()	Return the clustering coefficient for each vertex in nbunch
cluster_transitivity()	Return the transitivity (fraction of transitive triangles) of the graph.
szeged_index()	Return the Szeged index of the graph.

Automorphism group:

coarsest_equitable_refinement()	Return the coarsest partition which is finer than the input partition, and equitable with respect to self.
automorphism_group()	Return the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than the partition given.
is_vertex_transitive()	Return whether the automorphism group of self is transitive within the partition provided
is_isomorphic()	Test for isomorphism between self and other.
canonical_label()	Return the unique graph on \(\{0,1,...,n-1\}\) ( n = self.order() ) which 1) is isomorphic to self 2) is invariant in the isomorphism class.
is_cayley()	Check whether the graph is a Cayley graph.

Graph properties:

is_eulerian()	Return True if the graph has a (closed) tour that visits each edge exactly once.
is_planar()	Test whether the graph is planar.
is_circular_planar()	Test whether the graph is circular planar (outerplanar)
is_regular()	Return True if this graph is (\(k\)-)regular.
is_chordal()	Test whether the given graph is chordal.
is_circulant()	Test whether the graph is a circulant graph.
is_interval()	Check whether self is an interval graph
is_gallai_tree()	Return whether the current graph is a Gallai tree.
is_clique()	Test whether a set of vertices is a clique
is_cycle()	Test whether self is a (directed) cycle graph.
is_independent_set()	Test whether a set of vertices is an independent set
is_transitively_reduced()	Test whether the digraph is transitively reduced.
is_equitable()	Check whether the given partition is equitable with respect to self.

Traversals:

breadth_first_search()	Return an iterator over the vertices in a breadth-first ordering.
depth_first_search()	Return an iterator over the vertices in a depth-first ordering.
lex_BFS()	Perform a Lex BFS on the graph.

Distances:

centrality_betweenness()	Return the betweenness centrality
centrality_closeness()	Returns the closeness centrality (1/average distance to all vertices)
distance()	Return the (directed) distance from u to v in the (di)graph
distance_all_pairs()	Return the distances between all pairs of vertices.
distances_distribution()	Return the distances distribution of the (di)graph in a dictionary.
eccentricity()	Return the eccentricity of vertex (or vertices) v.
radius()	Return the radius of the (di)graph.
center()	Return the set of vertices in the center of the graph
diameter()	Return the largest distance between any two vertices.
distance_graph()	Return the graph on the same vertex set as the original graph but vertices are adjacent in the returned graph if and only if they are at specified distances in the original graph.
girth()	Compute the girth of the graph.
periphery()	Return the set of vertices in the periphery
shortest_path()	Return a list of vertices representing some shortest path from \(u\) to \(v\)
shortest_path_length()	Return the minimal length of paths from u to v
shortest_paths()	Return a dictionary associating to each vertex v a shortest path from u to v, if it exists.
shortest_path_lengths()	Return a dictionary of shortest path lengths keyed by targets that are connected by a path from u.
shortest_path_all_pairs()	Compute a shortest path between each pair of vertices.
wiener_index()	Return the Wiener index of the graph.
average_distance()	Return the average distance between vertices of the graph.

Flows, connectivity, trees:

is_connected()	Test whether the (di)graph is connected.
connected_components()	Return the list of connected components
connected_components_number()	Return the number of connected components.
connected_components_subgraphs()	Return a list of connected components as graph objects.
connected_component_containing_vertex()	Return a list of the vertices connected to vertex.
connected_components_sizes()	Return the sizes of the connected components as a list.
blocks_and_cut_vertices()	Compute the blocks and cut vertices of the graph.
blocks_and_cuts_tree()	Compute the blocks-and-cuts tree of the graph.
is_cut_edge()	Return True if the input edge is a cut-edge or a bridge.
is_cut_vertex()	Return True if the input vertex is a cut-vertex.
edge_cut()	Return a minimum edge cut between vertices \(s\) and \(t\)
vertex_cut()	Return a minimum vertex cut between non-adjacent vertices \(s\) and \(t\)
flow()	Return a maximum flow in the graph from x to y
edge_disjoint_paths()	Returns a list of edge-disjoint paths between two vertices
vertex_disjoint_paths()	Return a list of vertex-disjoint paths between two vertices as given by Menger’s theorem.
edge_connectivity()	Return the edge connectivity of the graph.
vertex_connectivity()	Return the vertex connectivity of the graph.
transitive_closure()	Compute the transitive closure of a graph and returns it.
transitive_reduction()	Return a transitive reduction of a graph.
min_spanning_tree()	Return the edges of a minimum spanning tree.
spanning_trees_count()	Return the number of spanning trees in a graph.
dominator_tree()	Returns a dominator tree of the graph.

Plot/embedding-related methods:

set_embedding()	Set a combinatorial embedding dictionary to _embedding attribute.
get_embedding()	Return the attribute _embedding if it exists.
faces()	Return the faces of an embedded graph.
get_pos()	Return the position dictionary
set_pos()	Set the position dictionary.
set_planar_positions()	Compute a planar layout for self using Schnyder’s algorithm
layout_planar()	Use Schnyder’s algorithm to compute a planar layout for self.
is_drawn_free_of_edge_crossings()	Test whether the position dictionary gives a planar embedding.
latex_options()	Return an instance of GraphLatex for the graph.
set_latex_options()	Set multiple options for rendering a graph with LaTeX.
layout()	Return a layout for the vertices of this graph.
layout_spring()	Compute a spring layout for this graph
layout_ranked()	Compute a ranked layout for this graph
layout_extend_randomly()	Extend randomly a partial layout
layout_circular()	Compute a circular layout for this graph
layout_tree()	Compute an ordered tree layout for this graph, which should be a tree (no non-oriented cycles).
layout_graphviz()	Call graphviz to compute a layout of the vertices of this graph.
graphplot()	Return a GraphPlot object.
plot()	Return a graphics object representing the (di)graph.
show()	Show the (di)graph.
plot3d()	Plot the graph in three dimensions.
show3d()	Plot the graph using Tachyon, and shows the resulting plot.
graphviz_string()	Return a representation in the dot language.
graphviz_to_file_named()	Write a representation in the dot in a file.

Algorithmically hard stuff:

steiner_tree()	Return a tree of minimum weight connecting the given set of vertices.
edge_disjoint_spanning_trees()	Return the desired number of edge-disjoint spanning trees/arborescences.
feedback_vertex_set()	Compute the minimum feedback vertex set of a (di)graph.
multiway_cut()	Return a minimum edge multiway cut
max_cut()	Return a maximum edge cut of the graph.
longest_path()	Return a longest path of self.
traveling_salesman_problem()	Solve the traveling salesman problem (TSP)
is_hamiltonian()	Test whether the current graph is Hamiltonian.
hamiltonian_cycle()	Return a Hamiltonian cycle/circuit of the current graph/digraph
hamiltonian_path()	Return a Hamiltonian path of the current graph/digraph
multicommodity_flow()	Solve a multicommodity flow problem.
disjoint_routed_paths()	Return a set of disjoint routed paths.
dominating_set()	Return a minimum dominating set of the graph
subgraph_search()	Return a copy of G in self.
subgraph_search_count()	Return the number of labelled occurences of G in self.
subgraph_search_iterator()	Return an iterator over the labelled copies of G in self.
characteristic_polynomial()	Return the characteristic polynomial of the adjacency matrix of the (di)graph.
genus()	Return the minimal genus of the graph.

Methods

class sage.graphs.generic_graph.GenericGraph

Bases: sage.graphs.generic_graph_pyx.GenericGraph_pyx

Base class for graphs and digraphs.

__eq__(other)

Compare self and other for equality.

Do not call this method directly. That is, for G.__eq__(H) write G == H.

Two graphs are considered equal if the following hold:

• they are either both directed, or both undirected;
• they have the same settings for loops, multiedges, and weightedness;
• they have the same set of vertices;
• they have the same (multi)set of arrows/edges, where labels of arrows/edges are taken into account if and only if the graphs are considered weighted. See weighted().

Note that this is not an isomorphism test.

EXAMPLES:

sage: G = graphs.EmptyGraph()
sage: H = Graph()
sage: G == H
True
sage: G.to_directed() == H.to_directed()
True
sage: G = graphs.RandomGNP(8,.9999)
sage: H = graphs.CompleteGraph(8)
sage: G == H # most often true
True
sage: G = Graph( {0:[1,2,3,4,5,6,7]} )
sage: H = Graph( {1:[0], 2:[0], 3:[0], 4:[0], 5:[0], 6:[0], 7:[0]} )
sage: G == H
True
sage: G.allow_loops(True)
sage: G == H
False
sage: G = graphs.RandomGNP(9,.3).to_directed()
sage: H = graphs.RandomGNP(9,.3).to_directed()
sage: G == H # most often false
False
sage: G = Graph(multiedges=True, sparse=True)
sage: G.add_edge(0,1)
sage: H = copy(G)
sage: H.add_edge(0,1)
sage: G == H
False

Note that graphs must be considered weighted, or Sage will not pay attention to edge label data in equality testing:

sage: foo = Graph(sparse=True)
sage: foo.add_edges([(0, 1, 1), (0, 2, 2)])
sage: bar = Graph(sparse=True)
sage: bar.add_edges([(0, 1, 2), (0, 2, 1)])
sage: foo == bar
True
sage: foo.weighted(True)
sage: foo == bar
False
sage: bar.weighted(True)
sage: foo == bar
False

add_clique(vertices, loops=False)

Add a clique to the graph with the given vertices.

If the vertices are already present, only the edges are added.

INPUT:

• vertices – an iterable with vertices for the clique to be added, e.g. a list, set, graph, etc.
• loops – (boolean, default: False) whether to add edges from every given vertex to itself. This is allowed only if the (di)graph allows loops.

EXAMPLES:

sage: G = Graph()
sage: G.add_clique(range(4))
sage: G.is_isomorphic(graphs.CompleteGraph(4))
True
sage: D = DiGraph()
sage: D.add_clique(range(4))
sage: D.is_isomorphic(digraphs.Complete(4))
True
sage: D = DiGraph(loops=True)
sage: D.add_clique(range(4), loops=True)
sage: D.is_isomorphic(digraphs.Complete(4, loops=True))
True
sage: D = DiGraph(loops=False)
sage: D.add_clique(range(4), loops=True)
Traceback (most recent call last):
...
ValueError: cannot add edge from 0 to 0 in graph without loops

If the list of vertices contains repeated elements, a loop will be added at that vertex, even if loops=False:

sage: G = Graph(loops=True)
sage: G.add_clique([1,1])
sage: G.edges()
[(1, 1, None)]

This is equivalent to:

sage: G = Graph(loops=True)
sage: G.add_clique([1], loops=True)
sage: G.edges()
[(1, 1, None)]

add_cycle(vertices)

Adds a cycle to the graph with the given vertices. If the vertices are already present, only the edges are added.

For digraphs, adds the directed cycle, whose orientation is determined by the list. Adds edges (vertices[u], vertices[u+1]) and (vertices[-1], vertices[0]).

INPUT:

• vertices – a list of indices for the vertices of the cycle to be added.

EXAMPLES:

sage: G = Graph()
sage: G.add_vertices(range(10)); G
Graph on 10 vertices
sage: show(G)
sage: G.add_cycle(list(range(10,20)))
sage: show(G)
sage: G.add_cycle(list(range(10)))
sage: show(G)

sage: D = DiGraph()
sage: D.add_cycle(list(range(4)))
sage: D.edges()
[(0, 1, None), (1, 2, None), (2, 3, None), (3, 0, None)]

add_edge(u, v=None, label=None)

Adds an edge from u and v.

INPUT: The following forms are all accepted:

• G.add_edge( 1, 2 )
• G.add_edge( (1, 2) )
• G.add_edges( [ (1, 2) ])
• G.add_edge( 1, 2, ‘label’ )
• G.add_edge( (1, 2, ‘label’) )
• G.add_edges( [ (1, 2, ‘label’) ] )

WARNING: The following intuitive input results in nonintuitive output:

sage: G = Graph()
sage: G.add_edge((1,2), 'label')
sage: G.networkx_graph().adj           # random output order
{'label': {(1, 2): None}, (1, 2): {'label': None}}

Use one of these instead:

sage: G = Graph()
sage: G.add_edge((1,2), label="label")
sage: G.networkx_graph().adj           # random output order
{1: {2: 'label'}, 2: {1: 'label'}}

sage: G = Graph()
sage: G.add_edge(1,2,'label')
sage: G.networkx_graph().adj           # random output order
{1: {2: 'label'}, 2: {1: 'label'}}

The following syntax is supported, but note that you must use the label keyword:

sage: G = Graph()
sage: G.add_edge((1,2), label='label')
sage: G.edges()
[(1, 2, 'label')]
sage: G = Graph()
sage: G.add_edge((1,2), 'label')
sage: G.edges()
[('label', (1, 2), None)]

Vertex name cannot be None, so:

sage: G = Graph()
sage: G.add_edge(None, 4)
sage: G.vertices()
[0, 4]

add_edges(edges, loops=True)

Add edges from an iterable container.

INPUT:

• edges – an iterable of edges, given either as (u, v) or (u, v, label).
• loops – (default: True) if False, remove all loops (v, v) from the input iterator. If None, remove loops unless the graph allows loops.

EXAMPLES:

sage: G = graphs.DodecahedralGraph()
sage: H = Graph()
sage: H.add_edges( G.edge_iterator() ); H
Graph on 20 vertices
sage: G = graphs.DodecahedralGraph().to_directed()
sage: H = DiGraph()
sage: H.add_edges( G.edge_iterator() ); H
Digraph on 20 vertices
sage: H.add_edges(iter([]))

sage: H = Graph()
sage: H.add_edges([(0, 1), (0, 2, "label")])
sage: H.edges()
[(0, 1, None), (0, 2, 'label')]

We demonstrate the loops argument:

sage: H = Graph()
sage: H.add_edges([(0,0)], loops=False); H.edges()
[]
sage: H.add_edges([(0,0)], loops=None); H.edges()
[]
sage: H.add_edges([(0,0)]); H.edges()
Traceback (most recent call last):
...
ValueError: cannot add edge from 0 to 0 in graph without loops
sage: H = Graph(loops=True)
sage: H.add_edges([(0,0)], loops=False); H.edges()
[]
sage: H.add_edges([(0,0)], loops=None); H.edges()
[(0, 0, None)]
sage: H.add_edges([(0,0)]); H.edges()
[(0, 0, None)]

add_path(vertices)

Adds a path to the graph with the given vertices. If the vertices are already present, only the edges are added.

For digraphs, adds the directed path vertices[0], ..., vertices[-1].

INPUT:

• vertices - a list of indices for the vertices of the path to be added.

EXAMPLES:

sage: G = Graph()
sage: G.add_vertices(range(10)); G
Graph on 10 vertices
sage: show(G)
sage: G.add_path(list(range(10,20)))
sage: show(G)
sage: G.add_path(list(range(10)))
sage: show(G)

sage: D = DiGraph()
sage: D.add_path(list(range(4)))
sage: D.edges()
[(0, 1, None), (1, 2, None), (2, 3, None)]

add_vertex(name=None)

Creates an isolated vertex. If the vertex already exists, then nothing is done.

INPUT:

• name - Name of the new vertex. If no name is specified, then the vertex will be represented by the least integer not already representing a vertex. Name must be an immutable object, and cannot be None.

As it is implemented now, if a graph \(G\) has a large number of vertices with numeric labels, then G.add_vertex() could potentially be slow, if name is None.

OUTPUT:

If name``=``None, the new vertex name is returned. None otherwise.

EXAMPLES:

sage: G = Graph(); G.add_vertex(); G
0
Graph on 1 vertex

sage: D = DiGraph(); D.add_vertex(); D
0
Digraph on 1 vertex

add_vertices(vertices)

Add vertices to the (di)graph from an iterable container of vertices. Vertices that already exist in the graph will not be added again.

INPUT:

• vertices: iterator of vertex labels. A new label is created, used and returned in the output list for all None values in vertices.

OUTPUT:

Generated names of new vertices if there is at least one None value present in vertices. None otherwise.

EXAMPLES:

sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7,8], 6: [8,9], 7: [9]}
sage: G = Graph(d)
sage: G.add_vertices([10,11,12])
sage: G.vertices()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
sage: G.add_vertices(graphs.CycleGraph(25).vertices())
sage: G.vertices()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]

sage: G = Graph()
sage: G.add_vertices([1,2,3])
sage: G.add_vertices([4,None,None,5])
[0, 6]

adjacency_matrix(sparse=None, vertices=None)

Returns the adjacency matrix of the (di)graph.

The matrix returned is over the integers. If a different ring is desired, use either sage.matrix.matrix0.Matrix.change_ring() method or matrix function.

INPUT:

• sparse - whether to represent with a sparse matrix
• vertices (list) – the ordering of the vertices defining how they should appear in the matrix. By default, the ordering given by GenericGraph.vertices() is used.

EXAMPLES:

sage: G = graphs.CubeGraph(4)
sage: G.adjacency_matrix()
[0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0]
[1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0]
[1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0]
[0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0]
[1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0]
[0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0]
[0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1]
[1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0]
[0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0]
[0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]

sage: matrix(GF(2),G) # matrix over GF(2)
[0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0]
[1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0]
[1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0]
[0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0]
[1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0]
[0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0]
[0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1]
[1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0]
[0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0]
[0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]

sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } )
sage: D.adjacency_matrix()
[0 1 1 1 0 0]
[1 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[1 0 0 0 0 1]
[0 1 0 0 0 0]

A different ordering of the vertices:

sage: graphs.PathGraph(5).adjacency_matrix(vertices=[2,4,1,3,0])
[0 0 1 1 0]
[0 0 0 1 0]
[1 0 0 0 1]
[1 1 0 0 0]
[0 0 1 0 0]

all_paths(start, end)

Returns a list of all paths (also lists) between a pair of vertices (start, end) in the (di)graph. If start is the same vertex as end, then [[start]] is returned – a list containing the 1-vertex, 0-edge path “start”.

EXAMPLES:

sage: eg1 = Graph({0:[1,2], 1:[4], 2:[3,4], 4:[5], 5:[6]})
sage: eg1.all_paths(0,6)
[[0, 1, 4, 5, 6], [0, 2, 4, 5, 6]]
sage: eg2 = graphs.PetersenGraph()
sage: sorted(eg2.all_paths(1,4))
[[1, 0, 4],
 [1, 0, 5, 7, 2, 3, 4],
 [1, 0, 5, 7, 2, 3, 8, 6, 9, 4],
 [1, 0, 5, 7, 9, 4],
 [1, 0, 5, 7, 9, 6, 8, 3, 4],
 [1, 0, 5, 8, 3, 2, 7, 9, 4],
 [1, 0, 5, 8, 3, 4],
 [1, 0, 5, 8, 6, 9, 4],
 [1, 0, 5, 8, 6, 9, 7, 2, 3, 4],
 [1, 2, 3, 4],
 [1, 2, 3, 8, 5, 0, 4],
 [1, 2, 3, 8, 5, 7, 9, 4],
 [1, 2, 3, 8, 6, 9, 4],
 [1, 2, 3, 8, 6, 9, 7, 5, 0, 4],
 [1, 2, 7, 5, 0, 4],
 [1, 2, 7, 5, 8, 3, 4],
 [1, 2, 7, 5, 8, 6, 9, 4],
 [1, 2, 7, 9, 4],
 [1, 2, 7, 9, 6, 8, 3, 4],
 [1, 2, 7, 9, 6, 8, 5, 0, 4],
 [1, 6, 8, 3, 2, 7, 5, 0, 4],
 [1, 6, 8, 3, 2, 7, 9, 4],
 [1, 6, 8, 3, 4],
 [1, 6, 8, 5, 0, 4],
 [1, 6, 8, 5, 7, 2, 3, 4],
 [1, 6, 8, 5, 7, 9, 4],
 [1, 6, 9, 4],
 [1, 6, 9, 7, 2, 3, 4],
 [1, 6, 9, 7, 2, 3, 8, 5, 0, 4],
 [1, 6, 9, 7, 5, 0, 4],
 [1, 6, 9, 7, 5, 8, 3, 4]]
sage: dg = DiGraph({0:[1,3], 1:[3], 2:[0,3]})
sage: sorted(dg.all_paths(0,3))
[[0, 1, 3], [0, 3]]
sage: ug = dg.to_undirected()
sage: sorted(ug.all_paths(0,3))
[[0, 1, 3], [0, 2, 3], [0, 3]]

Starting and ending at the same vertex (see trac ticket #13006):

sage: graphs.CompleteGraph(4).all_paths(2,2)
[[2]]

allow_loops(new, check=True)

Change whether loops are permitted in the (di)graph

INPUT:

• new - boolean.

EXAMPLES:

sage: G = Graph(loops=True); G
Looped graph on 0 vertices
sage: G.has_loops()
False
sage: G.allows_loops()
True
sage: G.add_edge((0,0))
sage: G.has_loops()
True
sage: G.loops()
[(0, 0, None)]
sage: G.allow_loops(False); G
Graph on 1 vertex
sage: G.has_loops()
False
sage: G.edges()
[]

sage: D = DiGraph(loops=True); D
Looped digraph on 0 vertices
sage: D.has_loops()
False
sage: D.allows_loops()
True
sage: D.add_edge((0,0))
sage: D.has_loops()
True
sage: D.loops()
[(0, 0, None)]
sage: D.allow_loops(False); D
Digraph on 1 vertex
sage: D.has_loops()
False
sage: D.edges()
[]

allow_multiple_edges(new, check=True, keep_label='any')

Changes whether multiple edges are permitted in the (di)graph.

INPUT:

• new (boolean): if True, the new graph will allow multiple edges.
• check (boolean): if True and new is False, we remove all multiple edges from the graph.
• keep_label ('any','min','max'): used only if new is False and check is True. If there are multiple edges with different labels, this variable defines which label should be kept: any label ('any'), the smallest label ('min'), or the largest ('max').

EXAMPLES:

The standard behavior with undirected graphs:

sage: G = Graph(multiedges=True,sparse=True); G
Multi-graph on 0 vertices
sage: G.has_multiple_edges()
False
sage: G.allows_multiple_edges()
True
sage: G.add_edges([(0,1,1), (0,1,2), (0,1,3)])
sage: G.has_multiple_edges()
True
sage: G.multiple_edges()
[(0, 1, 1), (0, 1, 2), (0, 1, 3)]
sage: G.allow_multiple_edges(False); G
Graph on 2 vertices
sage: G.has_multiple_edges()
False
sage: G.edges()
[(0, 1, 1)]

If we ask for the minimum label:

sage: G = Graph([(0, 1, 1), (0, 1, 2), (0, 1, 3)], multiedges=True,sparse=True)
sage: G.allow_multiple_edges(False, keep_label='min')
sage: G.edges()
[(0, 1, 1)]

If we ask for the maximum label:

sage: G = Graph([(0, 1, 1), (0, 1, 2), (0, 1, 3)], multiedges=True,sparse=True)
sage: G.allow_multiple_edges(False, keep_label='max')
sage: G.edges()
[(0, 1, 3)]

The standard behavior with digraphs:

sage: D = DiGraph(multiedges=True,sparse=True); D
Multi-digraph on 0 vertices
sage: D.has_multiple_edges()
False
sage: D.allows_multiple_edges()
True
sage: D.add_edges([(0,1)]*3)
sage: D.has_multiple_edges()
True
sage: D.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: D.allow_multiple_edges(False); D
Digraph on 2 vertices
sage: D.has_multiple_edges()
False
sage: D.edges()
[(0, 1, None)]

allows_loops()

Return whether loops are permitted in the (di)graph

EXAMPLES:

sage: G = Graph(loops=True); G
Looped graph on 0 vertices
sage: G.has_loops()
False
sage: G.allows_loops()
True
sage: G.add_edge((0,0))
sage: G.has_loops()
True
sage: G.loops()
[(0, 0, None)]
sage: G.allow_loops(False); G
Graph on 1 vertex
sage: G.has_loops()
False
sage: G.edges()
[]

sage: D = DiGraph(loops=True); D
Looped digraph on 0 vertices
sage: D.has_loops()
False
sage: D.allows_loops()
True
sage: D.add_edge((0,0))
sage: D.has_loops()
True
sage: D.loops()
[(0, 0, None)]
sage: D.allow_loops(False); D
Digraph on 1 vertex
sage: D.has_loops()
False
sage: D.edges()
[]

allows_multiple_edges()

Returns whether multiple edges are permitted in the (di)graph.

EXAMPLES:

sage: G = Graph(multiedges=True,sparse=True); G
Multi-graph on 0 vertices
sage: G.has_multiple_edges()
False
sage: G.allows_multiple_edges()
True
sage: G.add_edges([(0,1)]*3)
sage: G.has_multiple_edges()
True
sage: G.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: G.allow_multiple_edges(False); G
Graph on 2 vertices
sage: G.has_multiple_edges()
False
sage: G.edges()
[(0, 1, None)]

sage: D = DiGraph(multiedges=True,sparse=True); D
Multi-digraph on 0 vertices
sage: D.has_multiple_edges()
False
sage: D.allows_multiple_edges()
True
sage: D.add_edges([(0,1)]*3)
sage: D.has_multiple_edges()
True
sage: D.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: D.allow_multiple_edges(False); D
Digraph on 2 vertices
sage: D.has_multiple_edges()
False
sage: D.edges()
[(0, 1, None)]

am(sparse=None, vertices=None)

Returns the adjacency matrix of the (di)graph.

The matrix returned is over the integers. If a different ring is desired, use either sage.matrix.matrix0.Matrix.change_ring() method or matrix function.

INPUT:

• sparse - whether to represent with a sparse matrix
• vertices (list) – the ordering of the vertices defining how they should appear in the matrix. By default, the ordering given by GenericGraph.vertices() is used.

EXAMPLES:

sage: G = graphs.CubeGraph(4)
sage: G.adjacency_matrix()
[0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0]
[1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0]
[1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0]
[0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0]
[1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0]
[0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0]
[0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1]
[1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0]
[0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0]
[0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]

sage: matrix(GF(2),G) # matrix over GF(2)
[0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0]
[1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0]
[1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0]
[0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0]
[1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0]
[0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0]
[0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0]
[0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1]
[1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0]
[0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0]
[0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0]
[0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1]
[0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1]
[0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]

sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } )
sage: D.adjacency_matrix()
[0 1 1 1 0 0]
[1 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[1 0 0 0 0 1]
[0 1 0 0 0 0]

A different ordering of the vertices:

sage: graphs.PathGraph(5).adjacency_matrix(vertices=[2,4,1,3,0])
[0 0 1 1 0]
[0 0 0 1 0]
[1 0 0 0 1]
[1 1 0 0 0]
[0 0 1 0 0]

antisymmetric()

Tests whether the graph is antisymmetric.

A graph represents an antisymmetric relation if there being a path from a vertex x to a vertex y implies that there is not a path from y to x unless x=y.

A directed acyclic graph is antisymmetric. An undirected graph is never antisymmetric unless it is just a union of isolated vertices.

sage: graphs.RandomGNP(20,0.5).antisymmetric()
False
sage: digraphs.RandomDirectedGNR(20,0.5).antisymmetric()
True

automorphism_group(partition=None, verbosity=0, edge_labels=False, order=False, return_group=True, orbits=False, algorithm=None)

Return the automorphism group of the graph.

With partition this can also return the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than the partition given.

INPUT:

• partition - default is the unit partition, otherwise computes the subgroup of the full automorphism group respecting the partition.
• edge_labels - default False, otherwise allows only permutations respecting edge labels. Note that this option is not supported if algorithm="bliss"
• order - (default False) if True, compute the order of the automorphism group
• return_group - default True
• orbits - returns the orbits of the group acting on the vertices of the graph
• algorithm - If algorithm = "bliss" the automorphism group is computed using the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html). Setting it to “sage” uses Sage’s implementation. If set to None (default), bliss is used when available.

Warning

Since trac ticket #14319 the domain of the automorphism group is equal to the graph’s vertex set, and the translation argument has become useless.

OUTPUT: The order of the output is group, order, orbits. However, there are options to turn each of these on or off.

EXAMPLES:

Graphs:

sage: graphs_query = GraphQuery(display_cols=['graph6'],num_vertices=4)
sage: L = graphs_query.get_graphs_list()
sage: graphs_list.show_graphs(L)
sage: for g in L:
....:     G = g.automorphism_group()
....:     G.order(), G.gens()
(24, [(2,3), (1,2), (0,1)])
(4, [(2,3), (0,1)])
(2, [(1,2)])
(6, [(1,2), (0,1)])
(6, [(2,3), (1,2)])
(8, [(1,2), (0,1)(2,3)])
(2, [(0,1)(2,3)])
(2, [(1,2)])
(8, [(2,3), (0,1), (0,2)(1,3)])
(4, [(2,3), (0,1)])
(24, [(2,3), (1,2), (0,1)])
sage: C = graphs.CubeGraph(4)
sage: G = C.automorphism_group()
sage: M = G.character_table() # random order of rows, thus abs() below
sage: QQ(M.determinant()).abs()
712483534798848
sage: G.order()
384

sage: D = graphs.DodecahedralGraph()
sage: G = D.automorphism_group()
sage: A5 = AlternatingGroup(5)
sage: Z2 = CyclicPermutationGroup(2)
sage: H = A5.direct_product(Z2)[0] #see documentation for direct_product to explain the [0]
sage: G.is_isomorphic(H)
True

Multigraphs:

sage: G = Graph(multiedges=True,sparse=True)
sage: G.add_edge(('a', 'b'))
sage: G.add_edge(('a', 'b'))
sage: G.add_edge(('a', 'b'))
sage: G.automorphism_group()
Permutation Group with generators [('a','b')]

Digraphs:

sage: D = DiGraph( { 0:[1], 1:[2], 2:[3], 3:[4], 4:[0] } )
sage: D.automorphism_group()
Permutation Group with generators [(0,1,2,3,4)]

Edge labeled graphs:

sage: G = Graph(sparse=True)
sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] )
sage: G.automorphism_group(edge_labels=True)
Permutation Group with generators [(1,4)(2,3)]

sage: G = Graph({0 : {1 : 7}})
sage: G.automorphism_group(edge_labels=True)
Permutation Group with generators [(0,1)]

sage: foo = Graph(sparse=True)
sage: bar = Graph(implementation='c_graph',sparse=True)
sage: foo.add_edges([(0,1,1),(1,2,2), (2,3,3)])
sage: bar.add_edges([(0,1,1),(1,2,2), (2,3,3)])
sage: foo.automorphism_group(edge_labels=True)
Permutation Group with generators [()]
sage: foo.automorphism_group()
Permutation Group with generators [(0,3)(1,2)]
sage: bar.automorphism_group(edge_labels=True)
Permutation Group with generators [()]

You can also ask for just the order of the group:

sage: G = graphs.PetersenGraph()
sage: G.automorphism_group(return_group=False, order=True)
120

Or, just the orbits (note that each graph here is vertex transitive)

sage: G = graphs.PetersenGraph()
sage: G.automorphism_group(return_group=False, orbits=True,algorithm='sage')
[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]
sage: G.automorphism_group(partition=[[0],list(range(1,10))], return_group=False, orbits=True,algorithm='sage')
[[0], [2, 3, 6, 7, 8, 9], [1, 4, 5]]
sage: C = graphs.CubeGraph(3)
sage: C.automorphism_group(orbits=True, return_group=False,algorithm='sage')
[['000', '001', '010', '011', '100', '101', '110', '111']]

One can also use the faster algorithm for computing the automorphism group of the graph - bliss:

sage: G = graphs.HallJankoGraph()                   # optional - bliss
sage: A1 = G.automorphism_group()                   # optional - bliss
sage: A2 = G.automorphism_group(algorithm='bliss')  # optional - bliss
sage: A1.is_isomorphic(A2)                          # optional - bliss
True

average_degree()

Returns the average degree of the graph.

The average degree of a graph \(G=(V,E)\) is equal to \frac {2|E|}{|V|}.

EXAMPLES:

The average degree of a regular graph is equal to the degree of any vertex:

sage: g = graphs.CompleteGraph(5)
sage: g.average_degree() == 4
True

The average degree of a tree is always strictly less than \(2\):

sage: g = graphs.RandomGNP(20,.5)
sage: tree = Graph()
sage: tree.add_edges(g.min_spanning_tree())
sage: tree.average_degree() < 2
True

For any graph, it is equal to \frac {2|E|}{|V|}:

sage: g = graphs.RandomGNP(50,.8)
sage: g.average_degree() == 2*g.size()/g.order()
True

average_distance(by_weight=False, algorithm=None, weight_function=None)

Returns the average distance between vertices of the graph.

Formally, for a graph \(G\) this value is equal to \(\frac 1 {n(n-1)} \sum_{u,v\in G} d(u,v)\) where \(d(u,v)\) denotes the distance between vertices \(u\) and \(v\) and \(n\) is the number of vertices in \(G\).

For more information on the input variables and more examples, we refer to wiener_index() and shortest_path_all_pairs(), which have very similar input variables.

INPUT:

• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - the Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - the Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' for unweighted graphs, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost', otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

EXAMPLES:

From [GYLL93]:

sage: g=graphs.PathGraph(10)
sage: w=lambda x: (x*(x*x -1)/6)/(x*(x-1)/2)
sage: g.average_distance()==w(10)
True

REFERENCE:

[GYLL93]

I. Gutman, Y.-N. Yeh, S.-L. Lee, and Y.-L. Luo. Some recent results in the theory of the Wiener number. Indian Journal of Chemistry, 32A:651–661, 1993.

blocks_and_cut_vertices()

Computes the blocks and cut vertices of the graph.

In the case of a digraph, this computation is done on the underlying graph.

A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.

OUTPUT: ( B, C ), where B is a list of blocks- each is a list of vertices and the blocks are the corresponding induced subgraphs-and C is a list of cut vertices.

ALGORITHM:

We implement the algorithm proposed by Tarjan in [Tarjan72]. The original version is recursive. We emulate the recursion using a stack.

See also

• blocks_and_cuts_tree()
• is_biconnected()

EXAMPLES:

sage: graphs.PetersenGraph().blocks_and_cut_vertices()
([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], [])
sage: graphs.PathGraph(6).blocks_and_cut_vertices()
([[4, 5], [3, 4], [2, 3], [1, 2], [0, 1]], [1, 2, 3, 4])
sage: graphs.CycleGraph(7).blocks_and_cut_vertices()
([[0, 1, 2, 3, 4, 5, 6]], [])
sage: graphs.KrackhardtKiteGraph().blocks_and_cut_vertices()
([[8, 9], [7, 8], [0, 1, 2, 3, 4, 5, 6, 7]], [7, 8])
sage: G=Graph()  # make a bowtie graph where 0 is a cut vertex
sage: G.add_vertices(range(5))
sage: G.add_edges([(0,1),(0,2),(0,3),(0,4),(1,2),(3,4)])
sage: G.blocks_and_cut_vertices()
([[0, 1, 2], [0, 3, 4]], [0])
sage: graphs.StarGraph(3).blocks_and_cut_vertices()
([[0, 1], [0, 2], [0, 3]], [0])

REFERENCE:

[Tarjan72]

R.E. Tarjan. Depth-First Search and Linear Graph Algorithms. SIAM J. Comput. 1(2): 146-160 (1972).

blocks_and_cuts_tree()

Returns the blocks-and-cuts tree of self.

This new graph has two different kinds of vertices, some representing the blocks (type B) and some other the cut vertices of the graph self (type C).

There is an edge between a vertex \(u\) of type B and a vertex \(v\) of type C if the cut-vertex corresponding to \(v\) is in the block corresponding to \(u\).

The resulting graph is a tree, with the additional characteristic property that the distance between two leaves is even.

When self is biconnected, the tree is reduced to a single node of type \(B\).

See also

• blocks_and_cut_vertices()
• is_biconnected()

EXAMPLES:

sage: T = graphs.KrackhardtKiteGraph().blocks_and_cuts_tree(); T
Graph on 5 vertices
sage: T.is_isomorphic(graphs.PathGraph(5))
True

The distance between two leaves is even:

sage: T = graphs.RandomTree(40).blocks_and_cuts_tree()
sage: T.is_tree()
True
sage: leaves = [v for v in T if T.degree(v) == 1]
sage: all(T.distance(u,v) % 2 == 0 for u in leaves for v in leaves)
True

The tree of a biconnected graph has a single vertex, of type \(B\):

sage: T = graphs.PetersenGraph().blocks_and_cuts_tree()
sage: T.vertices()
[('B', (0, 1, 2, 3, 4, 5, 6, 7, 8, 9))]

REFERENCES:

[HarPri]

F. Harary and G. Prins. The block-cutpoint-tree of a graph. Publ. Math. Debrecen 13 1966 103-107.

[Gallai]

T. Gallai, Elementare Relationen bezueglich der Glieder und trennenden Punkte von Graphen, Magyar Tud. Akad. Mat. Kutato Int. Kozl. 9 (1964) 235-236

breadth_first_search(start, ignore_direction=False, distance=None, neighbors=None, report_distance=False)

Return an iterator over the vertices in a breadth-first ordering.

INPUT:

• start – vertex or list of vertices from which to start the traversal.
• ignore_direction – (default False) only applies to directed graphs. If True, searches across edges in either direction.
• distance – the maximum distance from the start nodes to traverse. The start nodes are distance zero from themselves.
• neighbors – a function giving the neighbors of a vertex. The function should take a vertex and return a list of vertices. For a graph, neighbors is by default the neighbors() function of the graph. For a digraph, the neighbors function defaults to the neighbor_out_iterator() function of the graph.
• report_distance – (default False) If True, reports pairs (vertex, distance) where distance is the distance from the start nodes. If False only the vertices are reported.

See also

• breadth_first_search – breadth-first search for fast compiled graphs.
• depth_first_search – depth-first search for fast compiled graphs.
• depth_first_search() – depth-first search for generic graphs.

EXAMPLES:

sage: G = Graph( { 0: [1], 1: [2], 2: [3], 3: [4], 4: [0]} )
sage: list(G.breadth_first_search(0))
[0, 1, 4, 2, 3]

By default, the edge direction of a digraph is respected, but this can be overridden by the ignore_direction parameter:

sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]})
sage: list(D.breadth_first_search(0))
[0, 1, 2, 3, 4, 5, 6, 7]
sage: list(D.breadth_first_search(0, ignore_direction=True))
[0, 1, 2, 3, 7, 4, 5, 6]

You can specify a maximum distance in which to search. A distance of zero returns the start vertices:

sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]})
sage: list(D.breadth_first_search(0,distance=0))
[0]
sage: list(D.breadth_first_search(0,distance=1))
[0, 1, 2, 3]

Multiple starting vertices can be specified in a list:

sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]})
sage: list(D.breadth_first_search([0]))
[0, 1, 2, 3, 4, 5, 6, 7]
sage: list(D.breadth_first_search([0,6]))
[0, 6, 1, 2, 3, 7, 4, 5]
sage: list(D.breadth_first_search([0,6],distance=0))
[0, 6]
sage: list(D.breadth_first_search([0,6],distance=1))
[0, 6, 1, 2, 3, 7]
sage: list(D.breadth_first_search(6,ignore_direction=True,distance=2))
[6, 3, 7, 0, 5]

More generally, you can specify a neighbors function. For example, you can traverse the graph backwards by setting neighbors to be the neighbors_in() function of the graph:

sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]})
sage: list(D.breadth_first_search(5,neighbors=D.neighbors_in, distance=2))
[5, 1, 2, 0]
sage: list(D.breadth_first_search(5,neighbors=D.neighbors_out, distance=2))
[5, 7, 0]
sage: list(D.breadth_first_search(5,neighbors=D.neighbors, distance=2))
[5, 1, 2, 7, 0, 4, 6]

It is possible (trac ticket #16470) using the keyword report_distance to get pairs (vertex, distance) encoding the distance to the starting vertices:

sage: G = graphs.PetersenGraph()
sage: list(G.breadth_first_search(0, report_distance=True))
[(0, 0), (1, 1), (4, 1), (5, 1), (2, 2), (6, 2), (3, 2), (9, 2),
(7, 2), (8, 2)]
sage: list(G.breadth_first_search(0, report_distance=False))
[0, 1, 4, 5, 2, 6, 3, 9, 7, 8]

sage: D = DiGraph({0:[1, 3], 1:[0, 2], 2:[0, 3], 3:[4]})
sage: D.show()
sage: list(D.breadth_first_search(4, neighbors=D.neighbor_in_iterator, report_distance=True))
[(4, 0), (3, 1), (0, 2), (2, 2), (1, 3)]

sage: C = graphs.CycleGraph(4)
sage: list(C.breadth_first_search([0,1], report_distance=True))
[(0, 0), (1, 0), (3, 1), (2, 1)]

canonical_label(partition=None, certificate=False, verbosity=0, edge_labels=False, algorithm=None, return_graph=True)

Return a graph on \(\{0,1,...,n-1\}\) ( n = self.order() ) which

• is isomorphic to self,
• is invariant in the isomorphism class.

In other words, given two graphs G and H which are isomorphic, suppose G_c and H_c are the graphs returned by canonical_label. Then the following hold:

• G_c == H_c
• G_c.adjacency_matrix() == H_c.adjacency_matrix()
• G_c.graph6_string() == H_c.graph6_string()

INPUT:

• partition - if given, the canonical label with respect to this set partition will be computed. The default is the unit set partition.

• certificate - if True, a dictionary mapping from the (di)graph to its canonical label will be given.

• verbosity - gets passed to nice: prints helpful output.

• edge_labels - default False, otherwise allows only permutations respecting edge labels.

• algorithm - If algorithm = "bliss" the automorphism group is computed using the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html). Setting it to “sage” uses Sage’s implementation. If set to None (default), bliss is used when available.

  Note

  Make sure you always compare canonical forms obtained by the same algorithm.

• return_graph - If return_graph = 'False' do not return the

  canonical graph.

EXAMPLES:

sage: D = graphs.DodecahedralGraph()
sage: E = D.canonical_label(algorithm='sage'); E
Dodecahedron: Graph on 20 vertices
sage: D.canonical_label(certificate=True, algorithm='sage')
(Dodecahedron: Graph on 20 vertices, {0: 0, 1: 19, 2: 16, 3: 15, 4: 9, 5: 1, 6: 10, 7: 8, 8: 14, 9: 12, 10: 17, 11: 11, 12: 5, 13: 6, 14: 2, 15: 4, 16: 3, 17: 7, 18: 13, 19: 18})
sage: D.is_isomorphic(E)
True

Multigraphs:

sage: G = Graph(multiedges=True,sparse=True)
sage: G.add_edge((0,1))
sage: G.add_edge((0,1))
sage: G.add_edge((0,1))
sage: G.canonical_label()
Multi-graph on 2 vertices
sage: Graph('A?', implementation='c_graph').canonical_label()
Graph on 2 vertices

Digraphs:

sage: P = graphs.PetersenGraph()
sage: DP = P.to_directed()
sage: DP.canonical_label(algorithm='sage').adjacency_matrix()
[0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 1 0 1 0 0 1]
[0 0 0 1 0 0 1 0 1 0]
[0 0 1 0 0 1 0 0 0 1]
[0 1 0 0 0 1 0 0 1 0]
[0 0 0 1 1 0 0 1 0 0]
[0 1 1 0 0 0 0 1 0 0]
[1 0 0 0 0 1 1 0 0 0]
[1 0 1 0 1 0 0 0 0 0]
[1 1 0 1 0 0 0 0 0 0]

Edge labeled graphs:

sage: G = Graph(sparse=True)
sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] )
sage: G.canonical_label(edge_labels=True)
Graph on 5 vertices
sage: G.canonical_label(edge_labels=True, certificate=True)
(Graph on 5 vertices, {0: 4, 1: 3, 2: 0, 3: 1, 4: 2})

Check for immutable graphs (trac ticket #16602):

sage: G = Graph([[1, 2], [2, 3]], immutable=True)
sage: C = G.canonical_label(); C
Graph on 3 vertices
sage: C.vertices()
[0, 1, 2]

Canonical forms can be computed by bliss as well:

sage: G = graphs.CubeGraph(6)                                       # optional - bliss
sage: H = G.copy()                                                  # optional - bliss
sage: s1 = G.canonical_label(return_graph=false,algorithm='bliss')  # optional - bliss
sage: s2 = H.canonical_label(return_graph=false,algorithm='bliss')  # optional - bliss
sage: s1 == s2                                                      # optional - bliss
True

cartesian_product(other)

Returns the Cartesian product of self and other.

The Cartesian product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)\) equal to the Cartesian product of the vertices \(V(G)\) and \(V(H)\), and \(((u,v), (w,x))\) is an edge iff either - \((u, w)\) is an edge of self and \(v = x\), or - \((v, x)\) is an edge of other and \(u = w\).

See also

• is_cartesian_product() – factorization of graphs according to the Cartesian product
• graph_products – a module on graph products.

categorical_product(other)

Returns the tensor product of self and other.

The tensor product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)\) equal to the Cartesian product of the vertices \(V(G)\) and \(V(H)\), and \(((u,v), (w,x))\) is an edge iff - \((u, w)\) is an edge of self, and - \((v, x)\) is an edge of other.

The tensor product is also known as the categorical product and the kronecker product (refering to the kronecker matrix product). See Wikipedia article on the Kronecker product.

EXAMPLES:

sage: Z = graphs.CompleteGraph(2)
sage: C = graphs.CycleGraph(5)
sage: T = C.tensor_product(Z); T
Graph on 10 vertices
sage: T.size()
10
sage: T.plot() # long time
Graphics object consisting of 21 graphics primitives

sage: D = graphs.DodecahedralGraph()
sage: P = graphs.PetersenGraph()
sage: T = D.tensor_product(P); T
Graph on 200 vertices
sage: T.size()
900
sage: T.plot() # long time
Graphics object consisting of 1101 graphics primitives

center(by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Returns the set of vertices in the center, i.e. whose eccentricity is equal to the radius of the (di)graph.

In other words, the center is the set of vertices achieving the minimum eccentricity.

For more information and examples on how to use input variables, see shortest_paths() and eccentricity()

INPUT:

• by_weight - if True, edge weights are taken into account; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - a Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' for unweighted graphs, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost', otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

EXAMPLES:

sage: G = graphs.DiamondGraph()
sage: G.center()
[1, 2]
sage: P = graphs.PetersenGraph()
sage: P.subgraph(P.center()) == P
True
sage: S = graphs.StarGraph(19)
sage: S.center()
[0]
sage: G = Graph()
sage: G.center()
[]
sage: G.add_vertex()
0
sage: G.center()
[0]

centrality_betweenness(k=None, normalized=True, weight=None, endpoints=False, seed=None, exact=False, algorithm=None)

Returns the betweenness centrality (fraction of number of shortest paths that go through each vertex) as a dictionary keyed by vertices. The betweenness is normalized by default to be in range (0,1).

Measures of the centrality of a vertex within a graph determine the relative importance of that vertex to its graph. Vertices that occur on more shortest paths between other vertices have higher betweenness than vertices that occur on less.

INPUT:

• normalized - boolean (default True) - if set to False, result is not normalized.
• k - integer or None (default None) - if set to an integer, use k node samples to estimate betweenness. Higher values give better approximations. Not available when algorithm="Sage".
• weight - None or string. If set to a string, use that attribute of the nodes as weight. weight = True is equivalent to weight = "weight". Not available when algorithm="Sage".
• endpoints - Boolean. If set to True it includes the endpoints in the shortest paths count. Not available when algorithm="Sage".
• exact (boolean, default: False) – whether to compute over rationals or on double C variables. Not available when algorithm="NetworkX".
• algorithm (default: None) – can be either "Sage" (see centrality), "NetworkX" or "None". In the latter case, Sage’s algorithm will be used whenever possible.

See also

• centrality_degree()
• centrality_closeness()

EXAMPLES:

sage: g = graphs.ChvatalGraph()
sage: g.centrality_betweenness() # abs tol 1e-10
{0: 0.06969696969696969, 1: 0.06969696969696969,
 2: 0.0606060606060606, 3: 0.0606060606060606,
 4: 0.06969696969696969, 5: 0.06969696969696969,
 6: 0.0606060606060606, 7: 0.0606060606060606,
 8: 0.0606060606060606, 9: 0.0606060606060606,
 10: 0.0606060606060606, 11: 0.0606060606060606}
sage: g.centrality_betweenness(normalized=False) # abs tol 1e-10
{0: 3.833333333333333, 1: 3.833333333333333, 2: 3.333333333333333,
 3: 3.333333333333333, 4: 3.833333333333333, 5: 3.833333333333333,
 6: 3.333333333333333, 7: 3.333333333333333, 8: 3.333333333333333,
 9: 3.333333333333333, 10: 3.333333333333333,
 11: 3.333333333333333}
sage: D = DiGraph({0:[1,2,3], 1:[2], 3:[0,1]})
sage: D.show(figsize=[2,2])
sage: D = D.to_undirected()
sage: D.show(figsize=[2,2])
sage: D.centrality_betweenness() # abs tol abs 1e-10
{0: 0.16666666666666666, 1: 0.16666666666666666, 2: 0.0, 3: 0.0}

centrality_closeness(vert=None, by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Returns the closeness centrality of all vertices in variable vert.

In a (strongly) connected graph, the closeness centrality of a vertex \(v\) is equal to the inverse of the average distance between \(v\) and other vertices. If the graph is disconnected, the closeness centrality of \(v\) is multiplied by the fraction of reachable vertices in the graph: this way, central vertices should also reach several other vertices in the graph [OLJ14]. In formulas,

\[c(v)=\frac{r(v)-1}{\sum_{w \in R(v)} d(v,w)}\frac{r(v)-1}{n-1}\]

where \(R(v)\) is the set of vertices reachable from \(v\), and \(r(v)\) is the cardinality of \(R(v)\).

‘Closeness centrality may be defined as the total graph-theoretic distance of a given vertex from all other vertices... Closeness is an inverse measure of centrality in that a larger value indicates a less central actor while a smaller value indicates a more central actor,’ [Borgatti95].

For more information, see the Wikipedia article Centrality.

INPUT:

• vert - the vertex or the list of vertices we want to analyze. If None (default), all vertices are considered.
• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS': performs a BFS from each vertex that has to be analyzed. Does not work with edge weights.
  • 'NetworkX': the NetworkX algorithm (works only with positive weights).
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Floyd-Warshall-Cython' - the Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False and all centralities are needed.
  • 'Floyd-Warshall-Python' - the Python implementation of the Floyd-Warshall algorithm. Works only if all centralities are needed, but it can deal with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' if by_weight is False, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost' otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

OUTPUT:

If vert is a vertex, the closeness centrality of that vertex. Otherwise, a dictionary associating to each vertex in vert its closeness centrality. If a vertex has (out)degree 0, its closeness centrality is not defined, and the vertex is not included in the output.

See also

• centrality_closeness_top_k()
• centrality_degree()
• centrality_betweenness()

REFERENCES:

[Borgatti95]

Stephen P. Borgatti. (1995). Centrality and AIDS. [Online] Available: http://www.analytictech.com/networks/centaids.htm

[OLJ14]

Paul W. Olsen, Alan G. Labouseur, Jeong-Hyon Hwang. Efficient Top-k Closeness Centrality Search Proceedings of the IEEE 30th International Conference on Data Engineering (ICDE), 2014

EXAMPLES:

Standard examples:

sage: (graphs.ChvatalGraph()).centrality_closeness()
{0: 0.61111111111111..., 1: 0.61111111111111..., 2: 0.61111111111111..., 3: 0.61111111111111..., 4: 0.61111111111111..., 5: 0.61111111111111..., 6: 0.61111111111111..., 7: 0.61111111111111..., 8: 0.61111111111111..., 9: 0.61111111111111..., 10: 0.61111111111111..., 11: 0.61111111111111...}
sage: D = DiGraph({0:[1,2,3], 1:[2], 3:[0,1]})
sage: D.show(figsize=[2,2])
sage: D.centrality_closeness(vert=[0,1])
{0: 1.0, 1: 0.3333333333333333}
sage: D = D.to_undirected()
sage: D.show(figsize=[2,2])
sage: D.centrality_closeness()
{0: 1.0, 1: 1.0, 2: 0.75, 3: 0.75}

In a (strongly) connected (di)graph, the closeness centrality of \(v\) is inverse of the average distance between \(v\) and all other vertices:

sage: g = graphs.PathGraph(5)
sage: g.centrality_closeness(0)
0.4
sage: dist = g.shortest_path_lengths(0).values()
sage: float(len(dist)-1) / sum(dist)
0.4
sage: d = g.to_directed()
sage: d.centrality_closeness(0)
0.4
sage: dist = d.shortest_path_lengths(0).values()
sage: float(len(dist)-1) / sum(dist)
0.4

If a vertex has (out)degree 0, its closeness centrality is not defined:

sage: g = Graph(5)
sage: g.centrality_closeness()
{}
sage: print(g.centrality_closeness(0))
None

Weighted graphs:

sage: D = graphs.GridGraph([2,2])
sage: weight_function = lambda e:10
sage: D.centrality_closeness([(0,0),(0,1)])                          # tol abs 1e-12
{(0, 0): 0.75, (0, 1): 0.75}
sage: D.centrality_closeness((0,0), weight_function=weight_function) # tol abs 1e-12
0.075

characteristic_polynomial(var='x', laplacian=False)

Returns the characteristic polynomial of the adjacency matrix of the (di)graph.

Let \(G\) be a (simple) graph with adjacency matrix \(A\). Let \(I\) be the identity matrix of dimensions the same as \(A\). The characteristic polynomial of \(G\) is defined as the determinant \(\det(xI - A)\).

Note

characteristic_polynomial and charpoly are aliases and thus provide exactly the same method.

INPUT:

• x – (default: 'x') the variable of the characteristic polynomial.
• laplacian – (default: False) if True, use the Laplacian matrix.

See also

• kirchhoff_matrix()
• laplacian_matrix()

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.characteristic_polynomial()
x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48
sage: P.charpoly()
x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48
sage: P.characteristic_polynomial(laplacian=True)
x^10 - 30*x^9 + 390*x^8 - 2880*x^7 + 13305*x^6 -
39882*x^5 + 77640*x^4 - 94800*x^3 + 66000*x^2 - 20000*x

charpoly(var='x', laplacian=False)

Returns the characteristic polynomial of the adjacency matrix of the (di)graph.

Let \(G\) be a (simple) graph with adjacency matrix \(A\). Let \(I\) be the identity matrix of dimensions the same as \(A\). The characteristic polynomial of \(G\) is defined as the determinant \(\det(xI - A)\).

Note

characteristic_polynomial and charpoly are aliases and thus provide exactly the same method.

INPUT:

• x – (default: 'x') the variable of the characteristic polynomial.
• laplacian – (default: False) if True, use the Laplacian matrix.

See also

• kirchhoff_matrix()
• laplacian_matrix()

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.characteristic_polynomial()
x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48
sage: P.charpoly()
x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48
sage: P.characteristic_polynomial(laplacian=True)
x^10 - 30*x^9 + 390*x^8 - 2880*x^7 + 13305*x^6 -
39882*x^5 + 77640*x^4 - 94800*x^3 + 66000*x^2 - 20000*x

clear()

Empties the graph of vertices and edges and removes name, associated objects, and position information.

EXAMPLES:

sage: G=graphs.CycleGraph(4); G.set_vertices({0:'vertex0'})
sage: G.order(); G.size()
4
4
sage: len(G._pos)
4
sage: G.name()
'Cycle graph'
sage: G.get_vertex(0)
'vertex0'
sage: H = G.copy(implementation='c_graph', sparse=True)
sage: H.clear()
sage: H.order(); H.size()
0
0
sage: len(H._pos)
0
sage: H.name()
''
sage: H.get_vertex(0)
sage: H = G.copy(implementation='c_graph', sparse=False)
sage: H.clear()
sage: H.order(); H.size()
0
0
sage: len(H._pos)
0
sage: H.name()
''
sage: H.get_vertex(0)

cluster_transitivity()

Returns the transitivity (fraction of transitive triangles) of the graph.

Transitivity is the fraction of all existing triangles and all connected triples (triads), \(T = 3\times\text{triangles} / \text{triads}\).

See also section “Clustering” in chapter “Algorithms” of [HSSNX].

EXAMPLES:

sage: (graphs.FruchtGraph()).cluster_transitivity()
0.25

cluster_triangles(nbunch=None, with_labels=False)

Returns the number of triangles for the set \(nbunch\) of vertices as a dictionary keyed by vertex.

See also section “Clustering” in chapter “Algorithms” of [HSSNX].

INPUT:

• nbunch - The vertices to inspect. If nbunch=None, returns data for all vertices in the graph.

REFERENCE:

[HSSNX]

(1, 2, 3, 4) Aric Hagberg, Dan Schult and Pieter Swart. NetworkX documentation. [Online] Available: http://networkx.github.io/documentation/latest/reference/index.html

EXAMPLES:

sage: (graphs.FruchtGraph()).cluster_triangles().values()
[1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0]
sage: (graphs.FruchtGraph()).cluster_triangles()
{0: 1, 1: 1, 2: 0, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 0, 9: 1, 10: 1, 11: 0}
sage: (graphs.FruchtGraph()).cluster_triangles(nbunch=[0,1,2])
{0: 1, 1: 1, 2: 0}

clustering_average(implementation=None)

Returns the average clustering coefficient.

The clustering coefficient of a node \(i\) is the fraction of existing triangles containing node \(i\) and all possible triangles containing \(i\): \(c_i = T(i) / \binom {k_i} 2\) where \(T(i)\) is the number of existing triangles through \(i\), and \(k_i\) is the degree of vertex \(i\).

A coefficient for the whole graph is the average of the \(c_i\).

See also section “Clustering” in chapter “Algorithms” of [HSSNX].

INPUT:

• implementation - one of 'boost', 'sparse_copy', 'dense_copy', 'networkx' or None (default). In the latter case, the best algorithm available is used. Note that only 'networkx' supports directed graphs.

EXAMPLES:

sage: (graphs.FruchtGraph()).clustering_average()
1/4
sage: (graphs.FruchtGraph()).clustering_average(implementation='networkx')
0.25

clustering_coeff(nodes=None, weight=False, implementation=None, return_vertex_weights=None)

Returns the clustering coefficient for each vertex in nodes as a dictionary keyed by vertex.

For an unweighted graph, the clustering coefficient of a node \(i\) is the fraction of existing triangles containing node \(i\) and all possible triangles containing \(i\): \(c_i = T(i) / \binom {k_i} 2\) where \(T(i)\) is the number of existing triangles through \(i\), and \(k_i\) is the degree of vertex \(i\).

For weighted graphs the clustering is defined as the geometric average of the subgraph edge weights, normalized by the maximum weight in the network.

The value of \(c_i\) is assigned \(0\) if \(k_i < 2\).

See also section “Clustering” in chapter “Algorithms” of [HSSNX].

INPUT:

• nodes - the vertices to inspect (default None, returns data on all vertices in graph)
• weight - string or boolean (default is False). If it is a string it used the indicated edge property as weight. weight = True is equivalent to weight = 'weight'
• implementation - one of 'boost', 'sparse_copy', 'dense_copy', 'networkx' or None (default). In the latter case, the best algorithm available is used. Note that only 'networkx' supports directed or weighted graphs, and that 'sparse_copy' and 'dense_copy' do not support node different from None

EXAMPLES:

sage: graphs.FruchtGraph().clustering_coeff()
{0: 1/3, 1: 1/3, 2: 0, 3: 1/3, 4: 1/3, 5: 1/3,
 6: 1/3, 7: 1/3, 8: 0, 9: 1/3, 10: 1/3, 11: 0}

sage: (graphs.FruchtGraph()).clustering_coeff(weight=True)
{0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0,
3: 0.3333333333333333, 4: 0.3333333333333333,
5: 0.3333333333333333, 6: 0.3333333333333333,
7: 0.3333333333333333, 8: 0.0, 9: 0.3333333333333333,
10: 0.3333333333333333, 11: 0.0}

sage: (graphs.FruchtGraph()).clustering_coeff(nodes=[0,1,2])
{0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0}

sage: (graphs.FruchtGraph()).clustering_coeff(nodes=[0,1,2],
....:   weight=True)
{0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0}

sage: (graphs.GridGraph([5,5])).clustering_coeff(nodes=[(0,0),(0,1),(2,2)])
{(0, 0): 0.0, (0, 1): 0.0, (2, 2): 0.0}

coarsest_equitable_refinement(partition, sparse=True)

Returns the coarsest partition which is finer than the input partition, and equitable with respect to self.

A partition is equitable with respect to a graph if for every pair of cells C1, C2 of the partition, the number of edges from a vertex of C1 to C2 is the same, over all vertices in C1.

A partition P1 is finer than P2 (P2 is coarser than P1) if every cell of P1 is a subset of a cell of P2.

INPUT:

• partition - a list of lists
• sparse - (default False) whether to use sparse or dense representation- for small graphs, use dense for speed

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.coarsest_equitable_refinement([[0],list(range(1,10))])
[[0], [2, 3, 6, 7, 8, 9], [1, 4, 5]]
sage: G = graphs.CubeGraph(3)
sage: verts = G.vertices()
sage: Pi = [verts[:1], verts[1:]]
sage: Pi
[['000'], ['001', '010', '011', '100', '101', '110', '111']]
sage: G.coarsest_equitable_refinement(Pi)
[['000'], ['011', '101', '110'], ['111'], ['001', '010', '100']]

Note that given an equitable partition, this function returns that partition:

sage: P = graphs.PetersenGraph()
sage: prt = [[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]]
sage: P.coarsest_equitable_refinement(prt)
[[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]]

sage: ss = (graphs.WheelGraph(6)).line_graph(labels=False)
sage: prt = [[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]
sage: ss.coarsest_equitable_refinement(prt)
Traceback (most recent call last):
...
TypeError: Partition ([[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]) is not valid for this graph: vertices are incorrect.

sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False)
sage: ss.coarsest_equitable_refinement(prt)
[[(0, 1)], [(1, 2), (1, 4)], [(0, 3)], [(0, 2), (0, 4)], [(2, 3), (3, 4)]]

ALGORITHM: Brendan D. McKay’s Master’s Thesis, University of Melbourne, 1976.

complement()

Returns the complement of the (di)graph.

The complement of a graph has the same vertices, but exactly those edges that are not in the original graph. This is not well defined for graphs with multiple edges.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.plot() # long time
Graphics object consisting of 26 graphics primitives
sage: PC = P.complement()
sage: PC.plot() # long time
Graphics object consisting of 41 graphics primitives

sage: graphs.TetrahedralGraph().complement().size()
0
sage: graphs.CycleGraph(4).complement().edges()
[(0, 2, None), (1, 3, None)]
sage: graphs.CycleGraph(4).complement()
complement(Cycle graph): Graph on 4 vertices
sage: G = Graph(multiedges=True, sparse=True)
sage: G.add_edges([(0,1)]*3)
sage: G.complement()
Traceback (most recent call last):
...
ValueError: This method is not known to work on graphs with
multiedges. Perhaps this method can be updated to handle them, but
in the meantime if you want to use it please disallow multiedges
using allow_multiple_edges().

connected_component_containing_vertex(vertex)

Returns a list of the vertices connected to vertex.

EXAMPLES:

sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: G.connected_component_containing_vertex(0)
[0, 1, 2, 3]
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: D.connected_component_containing_vertex(0)
[0, 1, 2, 3]

connected_components()

Returns the list of connected components.

Returns a list of lists of vertices, each list representing a connected component. The list is ordered from largest to smallest component.

EXAMPLES:

sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: G.connected_components()
[[0, 1, 2, 3], [4, 5, 6]]
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: D.connected_components()
[[0, 1, 2, 3], [4, 5, 6]]

connected_components_number()

Returns the number of connected components.

EXAMPLES:

sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: G.connected_components_number()
2
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: D.connected_components_number()
2

connected_components_sizes()

Return the sizes of the connected components as a list.

The list is sorted from largest to lower values.

EXAMPLES:

sage: for x in graphs(3):    print(x.connected_components_sizes())
[1, 1, 1]
[2, 1]
[3]
[3]

connected_components_subgraphs()

Returns a list of connected components as graph objects.

EXAMPLES:

sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: L = G.connected_components_subgraphs()
sage: graphs_list.show_graphs(L)
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: L = D.connected_components_subgraphs()
sage: graphs_list.show_graphs(L)

copy(weighted=None, implementation='c_graph', data_structure=None, sparse=None, immutable=None)

Change the graph implementation

INPUT:

• weighted boolean (default: None) – weightedness for the copy. Might change the equality class if not None.
• sparse (boolean) – sparse=True is an alias for data_structure="sparse", and sparse=False is an alias for data_structure="dense". Only used when implementation='c_graph' and data_structure=None.
• data_structure – one of "sparse", "static_sparse", or "dense". See the documentation of Graph or DiGraph. Only used when implementation='c_graph'.
• immutable (boolean) – whether to create a mutable/immutable copy. Only used when implementation='c_graph' and data_structure=None.
  • immutable=None (default) means that the graph and its copy will behave the same way.
  • immutable=True is a shortcut for data_structure='static_sparse' and implementation='c_graph'
  • immutable=False sets implementation to 'c_graph'. When immutable=False is used to copy an immutable graph, the data structure used is "sparse" unless anything else is specified.

Note

If the graph uses StaticSparseBackend and the _immutable flag, then self is returned rather than a copy (unless one of the optional arguments is used).

OUTPUT:

A Graph object.

Warning

Please use this method only if you need to copy but change the underlying implementation or weightedness. Otherwise simply do copy(g) instead of g.copy().

Warning

If weighted is passed and is not the weightedness of the original, then the copy will not equal the original.

EXAMPLES:

sage: g=Graph({0:[0,1,1,2]},loops=True,multiedges=True,sparse=True)
sage: g==copy(g)
True
sage: g=DiGraph({0:[0,1,1,2],1:[0,1]},loops=True,multiedges=True,sparse=True)
sage: g==copy(g)
True

Note that vertex associations are also kept:

sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() }
sage: T = graphs.TetrahedralGraph()
sage: T.set_vertices(d)
sage: T2 = copy(T)
sage: T2.get_vertex(0)
Dodecahedron: Graph on 20 vertices

Notice that the copy is at least as deep as the objects:

sage: T2.get_vertex(0) is T.get_vertex(0)
False

Examples of the keywords in use:

sage: G = graphs.CompleteGraph(19)
sage: H = G.copy(implementation='c_graph')
sage: H == G; H is G
True
False
sage: G1 = G.copy(sparse=True)
sage: G1==G
True
sage: G1 is G
False
sage: G2 = copy(G)
sage: G2 is G
False

Argument weighted affects the equality class:

sage: G = graphs.CompleteGraph(5)
sage: H1 = G.copy(weighted=False)
sage: H2 = G.copy(weighted=True)
sage: [G.weighted(), H1.weighted(), H2.weighted()]
[False, False, True]
sage: [G == H1, G == H2, H1 == H2]
[True, False, False]
sage: G.weighted(True)
sage: [G == H1, G == H2, H1 == H2]
[False, True, False]

cycle_basis(output='vertex')

Returns a list of cycles which form a basis of the cycle space of self.

A basis of cycles of a graph is a minimal collection of cycles (considered as sets of edges) such that the edge set of any cycle in the graph can be written as a \(Z/2Z\) sum of the cycles in the basis.

INPUT:

• output ('vertex' (default) or 'edge') – whether every cycle is given as a list of vertices or a list of edges.

OUTPUT:

A list of lists, each of them representing the vertices (or the edges) of a cycle in a basis.

ALGORITHM:

Uses the NetworkX library for graphs without multiple edges.

Otherwise, by the standard algorithm using a spanning tree.

EXAMPLES:

A cycle basis in Petersen’s Graph

sage: g = graphs.PetersenGraph()
sage: g.cycle_basis()
[[1, 2, 7, 5, 0], [8, 3, 2, 7, 5], [4, 3, 2, 7, 5, 0], [4, 9, 7, 5, 0], [8, 6, 9, 7, 5], [1, 6, 9, 7, 5, 0]]

One can also get the result as a list of lists of edges:

sage: g.cycle_basis(output='edge')
[[(1, 2, None), (2, 7, None), (7, 5, None), (5, 0, None),
(0, 1, None)], [(8, 3, None), (3, 2, None), (2, 7, None),
(7, 5, None), (5, 8, None)], [(4, 3, None), (3, 2, None),
(2, 7, None), (7, 5, None), (5, 0, None), (0, 4, None)],
[(4, 9, None), (9, 7, None), (7, 5, None), (5, 0, None),
(0, 4, None)], [(8, 6, None), (6, 9, None), (9, 7, None),
(7, 5, None), (5, 8, None)], [(1, 6, None), (6, 9, None),
(9, 7, None), (7, 5, None), (5, 0, None), (0, 1, None)]]

Checking the given cycles are algebraically free:

sage: g = graphs.RandomGNP(30,.4)
sage: basis = g.cycle_basis()

Building the space of (directed) edges over \(Z/2Z\). On the way, building a dictionary associating an unique vector to each undirected edge:

sage: m = g.size()
sage: edge_space = VectorSpace(FiniteField(2),m)
sage: edge_vector = dict( zip( g.edges(labels = False), edge_space.basis() ) )
sage: for (u,v),vec in edge_vector.items():
....:    edge_vector[(v,u)] = vec

Defining a lambda function associating a vector to the vertices of a cycle:

sage: vertices_to_edges = lambda x : zip( x, x[1:] + [x[0]] )
sage: cycle_to_vector = lambda x : sum( edge_vector[e] for e in vertices_to_edges(x) )

Finally checking the cycles are a free set:

sage: basis_as_vectors = [cycle_to_vector(_) for _ in basis]
sage: edge_space.span(basis_as_vectors).rank() == len(basis)
True

For undirected graphs with multiple edges:

sage: G = Graph([(0,2,'a'),(0,2,'b'),(0,1,'c'),(1,2,'d')], multiedges=True)
sage: G.cycle_basis()
[[0, 2], [2, 1, 0]]
sage: G.cycle_basis(output='edge')
[[(0, 2, 'a'), (2, 0, 'b')], [(2, 1, 'd'), (1, 0, 'c'),
(0, 2, 'a')]]

Disconnected graph:

sage: G.add_cycle(["Hey", "Wuuhuu", "Really ?"])
sage: G.cycle_basis()
[['Really ?', 'Hey', 'Wuuhuu'], [0, 2], [2, 1, 0]]
sage: G.cycle_basis(output='edge')
[[('Really ?', 'Hey', None),
  ('Hey', 'Wuuhuu', None),
  ('Wuuhuu', 'Really ?', None)],
 [(0, 2, 'a'), (2, 0, 'b')],
 [(2, 1, 'd'), (1, 0, 'c'), (0, 2, 'a')]]

Graph that allows multiple edges but does not contain any:

sage: G = graphs.CycleGraph(3)
sage: G.allow_multiple_edges(True)
sage: G.cycle_basis()
[[2, 1, 0]]

Not yet implemented for directed graphs with multiple edges:

sage: G = DiGraph([(0,2,'a'),(0,2,'b'),(0,1,'c'),(1,2,'d')], multiedges=True)
sage: G.cycle_basis()
Traceback (most recent call last):
...
NotImplementedError: not implemented for directed graphs
with multiple edges

degree(vertices=None, labels=False)

Gives the degree (in + out for digraphs) of a vertex or of vertices.

INPUT:

• vertices - If vertices is a single vertex, returns the number of neighbors of vertex. If vertices is an iterable container of vertices, returns a list of degrees. If vertices is None, same as listing all vertices.
• labels - see OUTPUT

OUTPUT: Single vertex- an integer. Multiple vertices- a list of integers. If labels is True, then returns a dictionary mapping each vertex to its degree.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.degree(5)
3

sage: K = graphs.CompleteGraph(9)
sage: K.degree()
[8, 8, 8, 8, 8, 8, 8, 8, 8]

sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } )
sage: D.degree(vertices = [0,1,2], labels=True)
{0: 5, 1: 4, 2: 3}
sage: D.degree()
[5, 4, 3, 3, 3, 2]

degree_histogram()

Return a list, whose ith entry is the frequency of degree i.

EXAMPLES:

sage: G = graphs.Grid2dGraph(9,12)
sage: G.degree_histogram()
[0, 0, 4, 34, 70]

sage: G = graphs.Grid2dGraph(9,12).to_directed()
sage: G.degree_histogram()
[0, 0, 0, 0, 4, 0, 34, 0, 70]

degree_iterator(vertices=None, labels=False)

Returns an iterator over the degrees of the (di)graph.

In the case of a digraph, the degree is defined as the sum of the in-degree and the out-degree, i.e. the total number of edges incident to a given vertex.

INPUT:

• labels (boolean) – if set to False (default) the method returns an iterator over degrees. Otherwise it returns an iterator over tuples (vertex, degree).
• vertices - if specified, restrict to this subset.

EXAMPLES:

sage: G = graphs.Grid2dGraph(3,4)
sage: for i in G.degree_iterator():
....:  print(i)
3
4
2
...
2
4
sage: for i in G.degree_iterator(labels=True):
....:  print(i)
((0, 1), 3)
((1, 2), 4)
((0, 0), 2)
...
((0, 3), 2)
((1, 1), 4)

sage: D = graphs.Grid2dGraph(2,4).to_directed()
sage: for i in D.degree_iterator():
....:  print(i)
6
6
...
4
6
sage: for i in D.degree_iterator(labels=True):
....:  print(i)
((0, 1), 6)
((1, 2), 6)
...
((1, 0), 4)
((0, 2), 6)

degree_sequence()

Return the degree sequence of this (di)graph.

EXAMPLES:

The degree sequence of an undirected graph:

sage: g = Graph({1: [2, 5], 2: [1, 5, 3, 4], 3: [2, 5], 4: [3], 5: [2, 3]})
sage: g.degree_sequence()
[4, 3, 3, 2, 2]

The degree sequence of a digraph:

sage: g = DiGraph({1: [2, 5, 6], 2: [3, 6], 3: [4, 6], 4: [6], 5: [4, 6]})
sage: g.degree_sequence()
[5, 3, 3, 3, 3, 3]

Degree sequences of some common graphs:

sage: graphs.PetersenGraph().degree_sequence()
[3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
sage: graphs.HouseGraph().degree_sequence()
[3, 3, 2, 2, 2]
sage: graphs.FlowerSnark().degree_sequence()
[3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]

degree_to_cell(vertex, cell)

Returns the number of edges from vertex to an edge in cell. In the case of a digraph, returns a tuple (in_degree, out_degree).

EXAMPLES:

sage: G = graphs.CubeGraph(3)
sage: cell = G.vertices()[:3]
sage: G.degree_to_cell('011', cell)
2
sage: G.degree_to_cell('111', cell)
0

sage: D = DiGraph({ 0:[1,2,3], 1:[3,4], 3:[4,5]})
sage: cell = [0,1,2]
sage: D.degree_to_cell(5, cell)
(0, 0)
sage: D.degree_to_cell(3, cell)
(2, 0)
sage: D.degree_to_cell(0, cell)
(0, 2)

delete_edge(u, v=None, label=None)

Delete the edge from u to v, returning silently if vertices or edge does not exist.

INPUT: The following forms are all accepted:

• G.delete_edge( 1, 2 )
• G.delete_edge( (1, 2) )
• G.delete_edges( [ (1, 2) ] )
• G.delete_edge( 1, 2, ‘label’ )
• G.delete_edge( (1, 2, ‘label’) )
• G.delete_edges( [ (1, 2, ‘label’) ] )

EXAMPLES:

sage: G = graphs.CompleteGraph(19).copy(implementation='c_graph')
sage: G.size()
171
sage: G.delete_edge( 1, 2 )
sage: G.delete_edge( (3, 4) )
sage: G.delete_edges( [ (5, 6), (7, 8) ] )
sage: G.size()
167

sage: G.delete_edge( 9, 10, 'label' )
sage: G.delete_edge( (11, 12, 'label') )
sage: G.delete_edges( [ (13, 14, 'label') ] )
sage: G.size()
167

sage: C = graphs.CompleteGraph(19).to_directed(sparse=True)
sage: C.size()
342
sage: C.delete_edge( 1, 2 )
sage: C.delete_edge( (3, 4) )
sage: C.delete_edges( [ (5, 6), (7, 8) ] )

sage: C.delete_edge( 9, 10, 'label' )
sage: C.delete_edge( (11, 12, 'label') )
sage: C.delete_edges( [ (13, 14, 'label') ] )
sage: C.size() # correct!
338
sage: C.has_edge( (11, 12) ) # correct!
True

delete_edges(edges)

Delete edges from an iterable container.

EXAMPLES:

sage: K12 = graphs.CompleteGraph(12)
sage: K4 = graphs.CompleteGraph(4)
sage: K12.size()
66
sage: K12.delete_edges(K4.edge_iterator())
sage: K12.size()
60

sage: K12 = graphs.CompleteGraph(12).to_directed()
sage: K4 = graphs.CompleteGraph(4).to_directed()
sage: K12.size()
132
sage: K12.delete_edges(K4.edge_iterator())
sage: K12.size()
120

delete_multiedge(u, v)

Deletes all edges from u and v.

EXAMPLES:

sage: G = Graph(multiedges=True,sparse=True)
sage: G.add_edges([(0,1), (0,1), (0,1), (1,2), (2,3)])
sage: G.edges()
[(0, 1, None), (0, 1, None), (0, 1, None), (1, 2, None), (2, 3, None)]
sage: G.delete_multiedge( 0, 1 )
sage: G.edges()
[(1, 2, None), (2, 3, None)]

sage: D = DiGraph(multiedges=True,sparse=True)
sage: D.add_edges([(0,1,1), (0,1,2), (0,1,3), (1,0,None), (1,2,None), (2,3,None)])
sage: D.edges()
[(0, 1, 1), (0, 1, 2), (0, 1, 3), (1, 0, None), (1, 2, None), (2, 3, None)]
sage: D.delete_multiedge( 0, 1 )
sage: D.edges()
[(1, 0, None), (1, 2, None), (2, 3, None)]

delete_vertex(vertex, in_order=False)

Deletes vertex, removing all incident edges. Deleting a non-existent vertex will raise an exception.

INPUT:

• in_order - (default False) If True, this deletes the ith vertex in the sorted list of vertices, i.e. G.vertices()[i]

EXAMPLES:

sage: G = Graph(graphs.WheelGraph(9))
sage: G.delete_vertex(0); G.show()

sage: D = DiGraph({0:[1,2,3,4,5],1:[2],2:[3],3:[4],4:[5],5:[1]})
sage: D.delete_vertex(0); D
Digraph on 5 vertices
sage: D.vertices()
[1, 2, 3, 4, 5]
sage: D.delete_vertex(0)
Traceback (most recent call last):
...
RuntimeError: Vertex (0) not in the graph.

sage: G = graphs.CompleteGraph(4).line_graph(labels=False)
sage: G.vertices()
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
sage: G.delete_vertex(0, in_order=True)
sage: G.vertices()
[(0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
sage: G = graphs.PathGraph(5)
sage: G.set_vertices({0: 'no delete', 1: 'delete'})
sage: G.delete_vertex(1)
sage: G.get_vertices()
{0: 'no delete', 2: None, 3: None, 4: None}
sage: G.get_pos()
{0: (0, 0), 2: (2, 0), 3: (3, 0), 4: (4, 0)}

delete_vertices(vertices)

Remove vertices from the (di)graph taken from an iterable container of vertices. Deleting a non-existent vertex will raise an exception.

EXAMPLES:

sage: D = DiGraph({0:[1,2,3,4,5],1:[2],2:[3],3:[4],4:[5],5:[1]})
sage: D.delete_vertices([1,2,3,4,5]); D
Digraph on 1 vertex
sage: D.vertices()
[0]
sage: D.delete_vertices([1])
Traceback (most recent call last):
...
RuntimeError: Vertex (1) not in the graph.

density()

Returns the density (number of edges divided by number of possible edges).

In the case of a multigraph, raises an error, since there is an infinite number of possible edges.

EXAMPLES:

sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7, 8], 6: [8,9], 7: [9]}
sage: G = Graph(d); G.density()
1/3
sage: G = Graph({0:[1,2], 1:[0] }); G.density()
2/3
sage: G = DiGraph({0:[1,2], 1:[0] }); G.density()
1/2

Note that there are more possible edges on a looped graph:

sage: G.allow_loops(True)
sage: G.density()
1/3

depth_first_search(start, ignore_direction=False, distance=None, neighbors=None)

Return an iterator over the vertices in a depth-first ordering.

INPUT:

• start - vertex or list of vertices from which to start the traversal
• ignore_direction - (default False) only applies to directed graphs. If True, searches across edges in either direction.
• distance - Deprecated. Broken, do not use.
• neighbors - a function giving the neighbors of a vertex. The function should take a vertex and return a list of vertices. For a graph, neighbors is by default the neighbors() function of the graph. For a digraph, the neighbors function defaults to the neighbor_out_iterator() function of the graph.

See also

• breadth_first_search()
• breadth_first_search – breadth-first search for fast compiled graphs.
• depth_first_search – depth-first search for fast compiled graphs.

EXAMPLES:

sage: G = Graph( { 0: [1], 1: [2], 2: [3], 3: [4], 4: [0]} )
sage: list(G.depth_first_search(0))
[0, 4, 3, 2, 1]

By default, the edge direction of a digraph is respected, but this can be overridden by the ignore_direction parameter:

sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]})
sage: list(D.depth_first_search(0))
[0, 3, 6, 7, 2, 5, 1, 4]
sage: list(D.depth_first_search(0, ignore_direction=True))
[0, 7, 6, 3, 5, 2, 1, 4]

Multiple starting vertices can be specified in a list:

sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]})
sage: list(D.depth_first_search([0]))
[0, 3, 6, 7, 2, 5, 1, 4]
sage: list(D.depth_first_search([0,6]))
[0, 3, 6, 7, 2, 5, 1, 4]

More generally, you can specify a neighbors function. For example, you can traverse the graph backwards by setting neighbors to be the neighbors_in() function of the graph:

sage: D = digraphs.Path(10)
sage: D.add_path([22,23,24,5])
sage: D.add_path([5,33,34,35])
sage: list(D.depth_first_search(5, neighbors=D.neighbors_in))
[5, 4, 3, 2, 1, 0, 24, 23, 22]
sage: list(D.breadth_first_search(5, neighbors=D.neighbors_in))
[5, 24, 4, 23, 3, 22, 2, 1, 0]
sage: list(D.depth_first_search(5, neighbors=D.neighbors_out))
[5, 6, 7, 8, 9, 33, 34, 35]
sage: list(D.breadth_first_search(5, neighbors=D.neighbors_out))
[5, 33, 6, 34, 7, 35, 8, 9]

diameter(by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Returns the diameter of the (di)graph.

The diameter is defined to be the maximum distance between two vertices. It is infinite if the (di)graph is not (strongly) connected.

For more information and examples on how to use input variables, see shortest_paths() and eccentricity()

INPUT:

• by_weight - if True, edge weights are taken into account; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - a Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'standard', '2sweep', 'multi-sweep', 'iFUB': these algorithms are implemented in sage.graphs.distances_all_pairs.diameter() They work only if by_weight==False. See the function documentation for more information.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'iFUB' if by_weight is False, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost', otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

EXAMPLES: The more symmetric a graph is, the smaller (diameter - radius) is.

sage: G = graphs.BarbellGraph(9, 3)
sage: G.radius()
3
sage: G.diameter()
6

sage: G = graphs.OctahedralGraph()
sage: G.radius()
2
sage: G.diameter()
2

disjoint_routed_paths(pairs, solver=None, verbose=0)

Returns a set of disjoint routed paths.

Given a set of pairs \((s_i,t_i)\), a set of disjoint routed paths is a set of \(s_i-t_i\) paths which can interset at their endpoints and are vertex-disjoint otherwise.

INPUT:

• pairs – list of pairs of vertices
• solver – Specify a Linear Program solver to be used. If set to None, the default one is used. function of MixedIntegerLinearProgram. See the documentation of MixedIntegerLinearProgram.solve for more informations.
• verbose (integer) – sets the level of verbosity. Set to \(0\) by default (quiet).

EXAMPLES:

Given a grid, finding two vertex-disjoint paths, the first one from the top-left corner to the bottom-left corner, and the second from the top-right corner to the bottom-right corner is easy

sage: g = graphs.GridGraph([5,5])
sage: p1,p2 = g.disjoint_routed_paths( [((0,0), (0,4)), ((4,4), (4,0))])

Though there is obviously no solution to the problem in which each corner is sending information to the opposite one:

sage: g = graphs.GridGraph([5,5])
sage: p1,p2 = g.disjoint_routed_paths( [((0,0), (4,4)), ((0,4), (4,0))])
Traceback (most recent call last):
...
EmptySetError: The disjoint routed paths do not exist.

disjoint_union(other, verbose_relabel=None, labels='pairs', immutable=None)

Return the disjoint union of self and other.

INPUT:

• verbose_relabel - deprecated.
• labels - (defaults to ‘pairs’) If set to ‘pairs’, each element v in the first graph will be named (0,v) and each element u in other will be named (1,u) in the result. If set to ‘integers’, the elements of the result will be relabeled with consecutive integers.
• immutable (boolean) – whether to create a mutable/immutable disjoint union. immutable=None (default) means that the graphs and their disjoint union will behave the same way.

See also

• union()
• join()

EXAMPLES:

sage: G = graphs.CycleGraph(3)
sage: H = graphs.CycleGraph(4)
sage: J = G.disjoint_union(H); J
Cycle graph disjoint_union Cycle graph: Graph on 7 vertices
sage: J.vertices()
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (1, 3)]
sage: J = G.disjoint_union(H, labels='integers'); J
Cycle graph disjoint_union Cycle graph: Graph on 7 vertices
sage: J.vertices()
[0, 1, 2, 3, 4, 5, 6]
sage: (G+H).vertices()  # '+'-operator is a shortcut
[0, 1, 2, 3, 4, 5, 6]

sage: G=Graph({'a': ['b']})
sage: G.name("Custom path")
sage: G.name()
'Custom path'
sage: H=graphs.CycleGraph(3)
sage: J=G.disjoint_union(H); J
Custom path disjoint_union Cycle graph: Graph on 5 vertices
sage: J.vertices()
[(0, 'a'), (0, 'b'), (1, 0), (1, 1), (1, 2)]

disjunctive_product(other)

Returns the disjunctive product of self and other.

The disjunctive product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)=V(G)\times V(H)\), and \(((u,v), (w,x))\) is an edge iff either :

• \((u, w)\) is an edge of \(G\), or
• \((v, x)\) is an edge of \(H\).

EXAMPLES:

sage: Z = graphs.CompleteGraph(2)
sage: D = Z.disjunctive_product(Z); D
Graph on 4 vertices
sage: D.plot() # long time
Graphics object consisting of 11 graphics primitives

sage: C = graphs.CycleGraph(5)
sage: D = C.disjunctive_product(Z); D
Graph on 10 vertices
sage: D.plot() # long time
Graphics object consisting of 46 graphics primitives

distance(u, v, by_weight=False)

Returns the (directed) distance from u to v in the (di)graph, i.e. the length of the shortest path from u to v.

This method simply calls shortest_path_length(), with default arguments. For more information, and for more option, we refer to that method.

INPUT:

• by_weight - if False, the graph is considered unweighted, and the distance is the number of edges in a shortest path. If True, the distance is the sum of edge labels (which are assumed to be numbers).

EXAMPLES:

sage: G = graphs.CycleGraph(9)
sage: G.distance(0,1)
1
sage: G.distance(0,4)
4
sage: G.distance(0,5)
4
sage: G = Graph({0:[], 1:[]})
sage: G.distance(0,1)
+Infinity
sage: G = Graph({ 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True)
sage: G.plot(edge_labels=True).show() # long time
sage: G.distance(0, 3)
2
sage: G.distance(0, 3, by_weight=True)
3

distance_all_pairs(by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Returns the distances between all pairs of vertices.

INPUT:

• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - the Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - the Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' if by_weight is False, 'Dijkstra_Boost' if all weights are positive, 'Floyd-Warshall-Cython' otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

OUTPUT:

A doubly indexed dictionary

Note

There is a Cython version of this method that is usually much faster for large graphs, as most of the time is actually spent building the final double dictionary. Everything on the subject is to be found in the distances_all_pairs module.

Note

This algorithm simply calls GenericGraph.shortest_path_all_pairs(), and we suggest to look at that method for more information and examples.

EXAMPLES:

The Petersen Graph:

sage: g = graphs.PetersenGraph()
sage: print(g.distance_all_pairs())
{0: {0: 0, 1: 1, 2: 2, 3: 2, 4: 1, 5: 1, 6: 2, 7: 2, 8: 2, 9: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 2, 5: 2, 6: 1, 7: 2, 8: 2, 9: 2}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 2, 5: 2, 6: 2, 7: 1, 8: 2, 9: 2}, 3: {0: 2, 1: 2, 2: 1, 3: 0, 4: 1, 5: 2, 6: 2, 7: 2, 8: 1, 9: 2}, 4: {0: 1, 1: 2, 2: 2, 3: 1, 4: 0, 5: 2, 6: 2, 7: 2, 8: 2, 9: 1}, 5: {0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 0, 6: 2, 7: 1, 8: 1, 9: 2}, 6: {0: 2, 1: 1, 2: 2, 3: 2, 4: 2, 5: 2, 6: 0, 7: 2, 8: 1, 9: 1}, 7: {0: 2, 1: 2, 2: 1, 3: 2, 4: 2, 5: 1, 6: 2, 7: 0, 8: 2, 9: 1}, 8: {0: 2, 1: 2, 2: 2, 3: 1, 4: 2, 5: 1, 6: 1, 7: 2, 8: 0, 9: 2}, 9: {0: 2, 1: 2, 2: 2, 3: 2, 4: 1, 5: 2, 6: 1, 7: 1, 8: 2, 9: 0}}

Testing on Random Graphs:

sage: g = graphs.RandomGNP(20,.3)
sage: distances = g.distance_all_pairs()
sage: all([g.distance(0,v) == distances[0][v] for v in g])
True

See also

• distance_matrix()
• shortest_path_all_pairs()

distance_graph(dist)

Returns the graph on the same vertex set as the original graph but vertices are adjacent in the returned graph if and only if they are at specified distances in the original graph.

INPUT:

• dist is a nonnegative integer or a list of nonnegative integers. Infinity may be used here to describe vertex pairs in separate components.

OUTPUT:

The returned value is an undirected graph. The vertex set is identical to the calling graph, but edges of the returned graph join vertices whose distance in the calling graph are present in the input dist. Loops will only be present if distance 0 is included. If the original graph has a position dictionary specifying locations of vertices for plotting, then this information is copied over to the distance graph. In some instances this layout may not be the best, and might even be confusing when edges run on top of each other due to symmetries chosen for the layout.

EXAMPLES:

sage: G = graphs.CompleteGraph(3)
sage: H = G.cartesian_product(graphs.CompleteGraph(2))
sage: K = H.distance_graph(2)
sage: K.am()
[0 0 0 1 0 1]
[0 0 1 0 1 0]
[0 1 0 0 0 1]
[1 0 0 0 1 0]
[0 1 0 1 0 0]
[1 0 1 0 0 0]

To obtain the graph where vertices are adjacent if their distance apart is d or less use a range() command to create the input, using d+1 as the input to range. Notice that this will include distance 0 and hence place a loop at each vertex. To avoid this, use range(1,d+1).

sage: G = graphs.OddGraph(4)
sage: d = G.diameter()
sage: n = G.num_verts()
sage: H = G.distance_graph(list(range(d+1)))
sage: H.is_isomorphic(graphs.CompleteGraph(n))
False
sage: H = G.distance_graph(list(range(1,d+1)))
sage: H.is_isomorphic(graphs.CompleteGraph(n))
True

A complete collection of distance graphs will have adjacency matrices that sum to the matrix of all ones.

sage: P = graphs.PathGraph(20)
sage: all_ones = sum([P.distance_graph(i).am() for i in range(20)])
sage: all_ones == matrix(ZZ, 20, 20, [1]*400)
True

Four-bit strings differing in one bit is the same as four-bit strings differing in three bits.

sage: G = graphs.CubeGraph(4)
sage: H = G.distance_graph(3)
sage: G.is_isomorphic(H)
True

The graph of eight-bit strings, adjacent if different in an odd number of bits.

sage: G = graphs.CubeGraph(8) # long time
sage: H = G.distance_graph([1,3,5,7]) # long time
sage: degrees = [0]*sum([binomial(8,j) for j in [1,3,5,7]]) # long time
sage: degrees.append(2^8) # long time
sage: degrees == H.degree_histogram() # long time
True

An example of using Infinity as the distance in a graph that is not connected.

sage: G = graphs.CompleteGraph(3)
sage: H = G.disjoint_union(graphs.CompleteGraph(2))
sage: L = H.distance_graph(Infinity)
sage: L.am()
[0 0 0 1 1]
[0 0 0 1 1]
[0 0 0 1 1]
[1 1 1 0 0]
[1 1 1 0 0]

AUTHOR:

Rob Beezer, 2009-11-25

distance_matrix()

Returns the distance matrix of the (strongly) connected (di)graph.

The distance matrix of a (strongly) connected (di)graph is a matrix whose rows and columns are indexed with the vertices of the (di) graph. The intersection of a row and column contains the respective distance between the vertices indexed at these position.

Warning

The ordering of vertices in the matrix has no reason to correspond to the order of vertices in vertices(). In particular, if two integers \(i,j\) are vertices of a graph \(G\) with distance matrix M, then M[i][i] is not necessarily the distance between vertices \(i\) and \(j\).

EXAMPLES:

sage: G = graphs.CubeGraph(3)
sage: G.distance_matrix()
[0 1 1 2 1 2 2 3]
[1 0 2 1 2 1 3 2]
[1 2 0 1 2 3 1 2]
[2 1 1 0 3 2 2 1]
[1 2 2 3 0 1 1 2]
[2 1 3 2 1 0 2 1]
[2 3 1 2 1 2 0 1]
[3 2 2 1 2 1 1 0]

The well known result of Graham and Pollak states that the determinant of the distance matrix of any tree of order n is (-1)^{n-1}(n-1)2^{n-2}

sage: all(T.distance_matrix().det() == (-1)^9*(9)*2^8 for T in graphs.trees(10))
True

See also

• distance_all_pairs() – computes the distance between any two vertices.

distances_distribution(G)

Returns the distances distribution of the (di)graph in a dictionary.

This method ignores all edge labels, so that the distance considered is the topological distance.

OUTPUT:

A dictionary d such that the number of pairs of vertices at distance k (if any) is equal to \(d[k] \cdot |V(G)| \cdot (|V(G)|-1)\).

Note

We consider that two vertices that do not belong to the same connected component are at infinite distance, and we do not take the trivial pairs of vertices \((v, v)\) at distance \(0\) into account. Empty (di)graphs and (di)graphs of order 1 have no paths and so we return the empty dictionary {}.

EXAMPLES:

An empty Graph:

sage: g = Graph()
sage: g.distances_distribution()
{}

A Graph of order 1:

sage: g = Graph()
sage: g.add_vertex(1)
sage: g.distances_distribution()
{}

A Graph of order 2 without edge:

sage: g = Graph()
sage: g.add_vertices([1,2])
sage: g.distances_distribution()
{+Infinity: 1}

The Petersen Graph:

sage: g = graphs.PetersenGraph()
sage: g.distances_distribution()
{1: 1/3, 2: 2/3}

A graph with multiple disconnected components:

sage: g = graphs.PetersenGraph()
sage: g.add_edge('good','wine')
sage: g.distances_distribution()
{1: 8/33, 2: 5/11, +Infinity: 10/33}

The de Bruijn digraph dB(2,3):

sage: D = digraphs.DeBruijn(2,3)
sage: D.distances_distribution()
{1: 1/4, 2: 11/28, 3: 5/14}

dominating_set(independent=False, total=False, value_only=False, solver=None, verbose=0)

Returns a minimum dominating set of the graph represented by the list of its vertices. For more information, see the Wikipedia article on dominating sets.

A minimum dominating set \(S\) of a graph \(G\) is a set of its vertices of minimal cardinality such that any vertex of \(G\) is in \(S\) or has one of its neighbors in \(S\).

As an optimization problem, it can be expressed as:

\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\forall v \in G, b_v+\sum_{(u,v)\in G.edges()} b_u\geq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]

INPUT:

• independent – boolean (default: False). If independent=True, computes a minimum independent dominating set.
• total – boolean (default: False). If total=True, computes a total dominating set.
• value_only – boolean (default: False)
  • If True, only the cardinality of a minimum dominating set is returned.
  • If False (default), a minimum dominating set is returned as the list of its vertices.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

EXAMPLES:

A basic illustration on a PappusGraph:

sage: g=graphs.PappusGraph()
sage: g.dominating_set(value_only=True)
5

If we build a graph from two disjoint stars, then link their centers we will find a difference between the cardinality of an independent set and a stable independent set:

sage: g = 2 * graphs.StarGraph(5)
sage: g.add_edge(0,6)
sage: len(g.dominating_set())
2
sage: len(g.dominating_set(independent=True))
6

The total dominating set of the Petersen graph has cardinality 4:

sage: G = graphs.PetersenGraph()
sage: G.dominating_set(total=True,value_only=True)
4

The dominating set is calculated for both the directed and undirected graphs (modification introduced in trac ticket #17905):

sage: g=digraphs.Path(3)
sage: g.dominating_set(value_only=True)
2
sage: g=graphs.PathGraph(3)
sage: g.dominating_set(value_only=True)
1

dominator_tree(g, root, return_dict=False)

Uses Boost to compute the dominator tree of g, rooted at root.

A node \(d\) dominates a node \(n\) if every path from the entry node root to \(n\) must go through \(d\). The immediate dominator of a node \(n\) is the unique node that strictly dominates \(n\) but does not dominate any other node that dominates \(n\). A dominator tree is a tree where each node’s children are those nodes it immediately dominates. For more information, see Wikipedia article Dominator_(graph_theory).

If the graph is connected and undirected, the parent of a vertex \(v\) is:

• the root if \(v\) is in the same biconnected component as the root;
• the first cut vertex in a path from \(v\) to the root, otherwise.

If the graph is not connected, the dominator tree of the whole graph is equal to the dominator tree of the connected component of the root.

If the graph is directed, computing a dominator tree is more complicated, and it needs time \(O(m\log m)\), where \(m\) is the number of edges. The implementation provided by Boost is the most general one, so it needs time \(O(m\log m)\) even for undirected graphs.

INPUT:

• g (generic_graph) - the input graph.
• root (vertex) - the root of the dominator tree.
• return_dict (boolean) - if True, the function returns a dictionary associating to each vertex its parent in the dominator tree. If False (default), it returns the whole tree, as a Graph or a DiGraph.

OUTPUT:

The dominator tree, as a graph or as a dictionary, depending on the value of return_dict. If the output is a dictionary, it will contain None in correspondence of root and of vertices that are not reachable from root. If the output is a graph, it will not contain vertices that are not reachable from root.

EXAMPLES:

An undirected grid is biconnected, and its dominator tree is a star (everyone’s parent is the root):

sage: g = graphs.GridGraph([2,2]).dominator_tree((0,0))
sage: g.to_dictionary()
{(0, 0): [(0, 1), (1, 0), (1, 1)], (0, 1): [(0, 0)], (1, 0): [(0, 0)], (1, 1): [(0, 0)]}

If the graph is made by two 3-cycles \(C_1,C_2\) connected by an edge \((v,w)\), with \(v \in C_1\), \(w \in C_2\), the cut vertices are \(v\) and \(w\), the biconnected components are \(C_1\), \(C_2\), and the edge \((v,w)\). If the root is in \(C_1\), the parent of each vertex in \(C_1\) is the root, the parent of \(w\) is \(v\), and the parent of each vertex in \(C_2\) is \(w\):

sage: G = 2 * graphs.CycleGraph(3)
sage: v = 0
sage: w = 3
sage: G.add_edge(v,w)
sage: G.dominator_tree(1, return_dict=True)
{0: 1, 1: None, 2: 1, 3: 0, 4: 3, 5: 3}

An example with a directed graph:

sage: g = digraphs.Circuit(10).dominator_tree(5)
sage: g.to_dictionary()
{0: [1], 1: [2], 2: [3], 3: [4], 4: [], 5: [6], 6: [7], 7: [8], 8: [9], 9: [0]}

If the output is a dictionary:

sage: graphs.GridGraph([2,2]).dominator_tree((0,0), return_dict = True)
{(0, 0): None, (0, 1): (0, 0), (1, 0): (0, 0), (1, 1): (0, 0)}

eccentricity(v=None, by_weight=False, algorithm=None, weight_function=None, check_weight=True, dist_dict=None, with_labels=False)

Return the eccentricity of vertex (or vertices) v.

The eccentricity of a vertex is the maximum distance to any other vertex.

For more information and examples on how to use input variables, see shortest_paths()

INPUT:

• v - either a single vertex or a list of vertices. If it is not specified, then it is taken to be all vertices.
• by_weight - if True, edge weights are taken into account; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - a Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False and v is None.
  • 'Floyd-Warshall-Python' - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed). However, v must be None.
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • 'From_Dictionary': uses the (already computed) distances, that are provided by input variable dist_dict.
  • None (default): Sage chooses the best algorithm: 'From_Dictionary' if dist_dict is not None, 'BFS' for unweighted graphs, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost' otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.
• dist_dict - used only if algorithm=='From_Dictionary' - a dict of dicts of distances.
• with_labels - Whether to return a list or a dict.

EXAMPLES:

sage: G = graphs.KrackhardtKiteGraph()
sage: G.eccentricity()
[4, 4, 4, 4, 4, 3, 3, 2, 3, 4]
sage: G.vertices()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: G.eccentricity(7)
2
sage: G.eccentricity([7,8,9])
[3, 4, 2]
sage: G.eccentricity([7,8,9], with_labels=True) == {8: 3, 9: 4, 7: 2}
True
sage: G = Graph( { 0 : [], 1 : [], 2 : [1] } )
sage: G.eccentricity()
[+Infinity, +Infinity, +Infinity]
sage: G = Graph({0:[]})
sage: G.eccentricity(with_labels=True)
{0: 0}
sage: G = Graph({0:[], 1:[]})
sage: G.eccentricity(with_labels=True)
{0: +Infinity, 1: +Infinity}
sage: G = Graph([(0,1,1), (1,2,1), (0,2,3)])
sage: G.eccentricity(algorithm = 'BFS')
[1, 1, 1]
sage: G.eccentricity(algorithm = 'Floyd-Warshall-Cython')
[1, 1, 1]
sage: G.eccentricity(by_weight = True, algorithm = 'Dijkstra_NetworkX')
[2, 1, 2]
sage: G.eccentricity(by_weight = True, algorithm = 'Dijkstra_Boost')
[2, 1, 2]
sage: G.eccentricity(by_weight = True, algorithm = 'Johnson_Boost')
[2, 1, 2]
sage: G.eccentricity(by_weight = True, algorithm = 'Floyd-Warshall-Python')
[2, 1, 2]
sage: G.eccentricity(dist_dict = G.shortest_path_all_pairs(by_weight = True)[0])
[2, 1, 2]

edge_boundary(vertices1, vertices2=None, labels=True, sort=True)

Returns a list of edges \((u,v,l)\) with \(u\) in vertices1 and \(v\) in vertices2. If vertices2 is None, then it is set to the complement of vertices1.

In a digraph, the external boundary of a vertex \(v\) are those vertices \(u\) with an arc \((v, u)\).

INPUT:

• labels - if False, each edge is a tuple \((u,v)\) of vertices.

EXAMPLES:

sage: K = graphs.CompleteBipartiteGraph(9,3)
sage: len(K.edge_boundary( [0,1,2,3,4,5,6,7,8], [9,10,11] ))
27
sage: K.size()
27

Note that the edge boundary preserves direction:

sage: K = graphs.CompleteBipartiteGraph(9,3).to_directed()
sage: len(K.edge_boundary( [0,1,2,3,4,5,6,7,8], [9,10,11] ))
27
sage: K.size()
54

sage: D = DiGraph({0:[1,2], 3:[0]})
sage: D.edge_boundary([0])
[(0, 1, None), (0, 2, None)]
sage: D.edge_boundary([0], labels=False)
[(0, 1), (0, 2)]

edge_connectivity(value_only=True, implementation=None, use_edge_labels=False, vertices=False, solver=None, verbose=0)

Returns the edge connectivity of the graph.

For more information, see the Wikipedia article on connectivity.

Note

When the graph is a directed graph, this method actually computes the strong connectivity, (i.e. a directed graph is strongly \(k\)-connected if there are \(k\) disjoint paths between any two vertices \(u, v\)). If you do not want to consider strong connectivity, the best is probably to convert your DiGraph object to a Graph object, and compute the connectivity of this other graph.

INPUT:

• value_only – boolean (default: True)
  • When set to True (default), only the value is returned.
  • When set to False, both the value and a minimum edge cut are returned.
• implementation – selects an implementation:
  • When set to None (default): selects the best implementation available.
  • When set to "boost", we use the Boost graph library (which is much more efficient). It is not available when edge_labels=True, and it is unreliable for directed graphs (see trac ticket #18753).
  • When set to "Sage", we use Sage’s implementation.
• use_edge_labels – boolean (default: False)
  • When set to True, computes a weighted minimum cut where each edge has a weight defined by its label. (If an edge has no label, \(1\) is assumed.). Implies boost = False.
  • When set to False, each edge has weight \(1\).
• vertices – boolean (default: False)
  • When set to True, also returns the two sets of vertices that are disconnected by the cut. Implies value_only=False.
• solver – (default: None) Specify a Linear Program (LP) solver to be used (ignored if implementation='boost'). If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

EXAMPLES:

A basic application on the PappusGraph:

sage: g = graphs.PappusGraph()
sage: g.edge_connectivity()
3

The edge connectivity of a complete graph ( and of a random graph ) is its minimum degree, and one of the two parts of the bipartition is reduced to only one vertex. The cutedges isomorphic to a Star graph:

sage: g = graphs.CompleteGraph(5)
sage: [ value, edges, [ setA, setB ]] = g.edge_connectivity(vertices=True)
sage: value
4
sage: len(setA) == 1 or len(setB) == 1
True
sage: cut = Graph()
sage: cut.add_edges(edges)
sage: cut.is_isomorphic(graphs.StarGraph(4))
True

Even if obviously in any graph we know that the edge connectivity is less than the minimum degree of the graph:

sage: g = graphs.RandomGNP(10,.3)
sage: min(g.degree()) >= g.edge_connectivity()
True

If we build a tree then assign to its edges a random value, the minimum cut will be the edge with minimum value:

sage: g = graphs.RandomGNP(15,.5)
sage: tree = Graph()
sage: tree.add_edges(g.min_spanning_tree())
sage: for u,v in tree.edge_iterator(labels=None):
....:      tree.set_edge_label(u,v,random())
sage: minimum = min([l for u,v,l in tree.edge_iterator()])
sage: [value, [(u,v,l)]] = tree.edge_connectivity(value_only=False, use_edge_labels=True)
sage: l == minimum
True

When value_only = True and implementation="sage", this function is optimized for small connectivity values and does not need to build a linear program.

It is the case for graphs which are not connected

sage: g = 2 * graphs.PetersenGraph()
sage: g.edge_connectivity(implementation="sage")
0.0

For directed graphs, the strong connectivity is tested through the dedicated function

sage: g = digraphs.ButterflyGraph(3)
sage: g.edge_connectivity(implementation="sage")
0.0

We check that the result with Boost is the same as the result without Boost

sage: g = graphs.RandomGNP(15,.3)
sage: g.edge_connectivity() == g.edge_connectivity(implementation="sage")
True

Boost interface also works with directed graphs

sage: digraphs.Circuit(10).edge_connectivity(implementation = "boost", vertices = True)
[1, [(0, 1)], [{0}, {1, 2, 3, 4, 5, 6, 7, 8, 9}]]

However, the Boost algorithm is not reliable if the input is directed (see trac ticket #18753):

sage: g = digraphs.Path(3)
sage: g.edge_connectivity()
0.0
sage: g.edge_connectivity(implementation="boost")
1
sage: g.add_edge(1,0)
sage: g.edge_connectivity()
0.0
sage: g.edge_connectivity(implementation="boost")
0

edge_cut(s, t, value_only=True, use_edge_labels=False, vertices=False, algorithm='FF', solver=None, verbose=0)

Return a minimum edge cut between vertices \(s\) and \(t\).

A minimum edge cut between two vertices \(s\) and \(t\) of self is a set \(A\) of edges of minimum weight such that the graph obtained by removing \(A\) from the graph is disconnected. For more information, see the Wikipedia article on cuts.

INPUT:

• s – source vertex

• t – sink vertex

• value_only – boolean (default: True). When set to True, only the weight of a minimum cut is returned. Otherwise, a list of edges of a minimum cut is also returned.

• use_edge_labels – boolean (default: False). When set to True, computes a weighted minimum cut where each edge has a weight defined by its label (if an edge has no label, \(1\) is assumed). Otherwise, each edge has weight \(1\).

• vertices – boolean (default: False). When set to True, returns a list of edges in the edge cut and the two sets of vertices that are disconnected by the cut.

  Note: vertices=True implies value_only=False.

• algorithm – algorithm to use:

  • If algorithm = "FF", a Python implementation of the Ford-Fulkerson algorithm is used
  • If algorithm = "LP", the flow problem is solved using Linear Programming.
  • If algorithm = "igraph", the igraph implementation of the Goldberg-Tarjan algorithm is used (only available when igraph is installed)
  • If algorithm = None (default), we use LP if vertex_bound = True, otherwise, we use igraph if it is available, FF if it is not available.

• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

Note

The use of Linear Programming for non-integer problems may possibly mean the presence of a (slight) numerical noise.

OUTPUT:

Real number or tuple, depending on the given arguments (examples are given below).

EXAMPLES:

A basic application in the Pappus graph:

sage: g = graphs.PappusGraph()
sage: g.edge_cut(1, 2, value_only=True)
3

Or on Petersen’s graph, with the corresponding bipartition of the vertex set:

sage: g = graphs.PetersenGraph()
sage: g.edge_cut(0, 3, vertices=True)
[3, [(0, 1, None), (0, 4, None), (0, 5, None)], [[0], [1, 2, 3, 4, 5, 6, 7, 8, 9]]]

If the graph is a path with randomly weighted edges:

sage: g = graphs.PathGraph(15)
sage: for (u,v) in g.edge_iterator(labels=None):
....:    g.set_edge_label(u,v,random())

The edge cut between the two ends is the edge of minimum weight:

sage: minimum = min([l for u,v,l in g.edge_iterator()])
sage: minimum == g.edge_cut(0, 14, use_edge_labels=True)
True
sage: [value,[e]] = g.edge_cut(0, 14, use_edge_labels=True, value_only=False)
sage: g.edge_label(e[0],e[1]) == minimum
True

The two sides of the edge cut are obviously shorter paths:

sage: value,edges,[set1,set2] = g.edge_cut(0, 14, use_edge_labels=True, vertices=True)
sage: g.subgraph(set1).is_isomorphic(graphs.PathGraph(len(set1)))
True
sage: g.subgraph(set2).is_isomorphic(graphs.PathGraph(len(set2)))
True
sage: len(set1) + len(set2) == g.order()
True

edge_disjoint_paths(s, t, algorithm='FF')

Returns a list of edge-disjoint paths between two vertices as given by Menger’s theorem.

The edge version of Menger’s theorem asserts that the size of the minimum edge cut between two vertices \(s\) and`t` (the minimum number of edges whose removal disconnects \(s\) and \(t\)) is equal to the maximum number of pairwise edge-independent paths from \(s\) to \(t\).

This function returns a list of such paths.

INPUT:

• algorithm – There are currently two different implementations of this method :

  • If algorithm = "FF" (default), a Python implementation of the Ford-Fulkerson algorithm is used.
  • If algorithm = "LP", the flow problem is solved using Linear Programming.

Note

This function is topological: it does not take the eventual weights of the edges into account.

EXAMPLES:

In a complete bipartite graph

sage: g = graphs.CompleteBipartiteGraph(2,3)
sage: g.edge_disjoint_paths(0,1)
[[0, 2, 1], [0, 3, 1], [0, 4, 1]]

edge_disjoint_spanning_trees(k, root=None, solver=None, verbose=0)

Returns the desired number of edge-disjoint spanning trees/arborescences.

INPUT:

• k (integer) – the required number of edge-disjoint spanning trees/arborescences
• root (vertex) – root of the disjoint arborescences when the graph is directed. If set to None, the first vertex in the graph is picked.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

ALGORITHM:

Mixed Integer Linear Program. The formulation can be found in [LPForm].

There are at least two possible rewritings of this method which do not use Linear Programming:

• The algorithm presented in the paper entitled “A short proof of the tree-packing theorem”, by Thomas Kaiser [KaisPacking].
• The implementation of a Matroid class and of the Matroid Union Theorem (see section 42.3 of [SchrijverCombOpt]), applied to the cycle Matroid (see chapter 51 of [SchrijverCombOpt]).

EXAMPLES:

The Petersen Graph does have a spanning tree (it is connected):

sage: g = graphs.PetersenGraph()
sage: [T] = g.edge_disjoint_spanning_trees(1)
sage: T.is_tree()
True

Though, it does not have 2 edge-disjoint trees (as it has less than \(2(|V|-1)\) edges):

sage: g.edge_disjoint_spanning_trees(2)
Traceback (most recent call last):
...
EmptySetError: This graph does not contain the required number of trees/arborescences !

By Edmond’s theorem, a graph which is \(k\)-connected always has \(k\) edge-disjoint arborescences, regardless of the root we pick:

sage: g = digraphs.RandomDirectedGNP(28,.3) # reduced from 30 to 28, cf. #9584
sage: k = Integer(g.edge_connectivity())
sage: arborescences = g.edge_disjoint_spanning_trees(k)  # long time (up to 15s on sage.math, 2011)
sage: all([a.is_directed_acyclic() for a in arborescences])  # long time
True
sage: all([a.is_connected() for a in arborescences])  # long time
True

In the undirected case, we can only ensure half of it:

sage: g = graphs.RandomGNP(30,.3)
sage: k = floor(Integer(g.edge_connectivity())/2)
sage: trees = g.edge_disjoint_spanning_trees(k)
sage: all([t.is_tree() for t in trees])
True

REFERENCES:

[LPForm]

Nathann Cohen, Several Graph problems and their Linear Program formulations, http://hal.archives-ouvertes.fr/inria-00504914/en

[KaisPacking]

Thomas Kaiser A short proof of the tree-packing theorem Arxiv 0911.2809

[SchrijverCombOpt]

(1, 2) Alexander Schrijver Combinatorial optimization: polyhedra and efficiency 2003

edge_iterator(vertices=None, labels=True, ignore_direction=False)

Returns an iterator over edges.

The iterator returned is over the edges incident with any vertex given in the parameter vertices. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only.

INPUT:

• vertices - (default: None) a vertex, a list of vertices or None
• labels - if False, each edge is a tuple (u,v) of vertices.
• ignore_direction - bool (default: False) - only applies to directed graphs. If True, searches across edges in either direction.

EXAMPLES:

sage: for i in graphs.PetersenGraph().edge_iterator([0]):
....:  print(i)
(0, 1, None)
(0, 4, None)
(0, 5, None)
sage: D = DiGraph( { 0 : [1,2], 1: [0] } )
sage: for i in D.edge_iterator([0]):
....:  print(i)
(0, 1, None)
(0, 2, None)

sage: G = graphs.TetrahedralGraph()
sage: list(G.edge_iterator(labels=False))
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]

sage: D = DiGraph({1:[0], 2:[0]})
sage: list(D.edge_iterator(0))
[]
sage: list(D.edge_iterator(0, ignore_direction=True))
[(1, 0, None), (2, 0, None)]

edge_label(u, v=None)

Returns the label of an edge. Note that if the graph allows multiple edges, then a list of labels on the edge is returned.

EXAMPLES:

sage: G = Graph({0 : {1 : 'edgelabel'}}, sparse=True)
sage: G.edges(labels=False)
[(0, 1)]
sage: G.edge_label( 0, 1 )
'edgelabel'
sage: D = DiGraph({0 : {1 : 'edgelabel'}}, sparse=True)
sage: D.edges(labels=False)
[(0, 1)]
sage: D.edge_label( 0, 1 )
'edgelabel'

sage: G = Graph(multiedges=True, sparse=True)
sage: [G.add_edge(0,1,i) for i in range(1,6)]
[None, None, None, None, None]
sage: sorted(G.edge_label(0,1))
[1, 2, 3, 4, 5]

edge_labels()

Return a list of edge labels.

EXAMPLES:

sage: G = Graph({0:{1:'x',2:'z',3:'a'}, 2:{5:'out'}}, sparse=True)
sage: G.edge_labels()
['x', 'z', 'a', 'out']
sage: G = DiGraph({0:{1:'x',2:'z',3:'a'}, 2:{5:'out'}}, sparse=True)
sage: G.edge_labels()
['x', 'z', 'a', 'out']

edges(labels=True, sort=True, key=None)

Return a list of edges.

Each edge is a triple (u,v,l) where u and v are vertices and l is a label. If the parameter labels is False then a list of couple (u,v) is returned where u and v are vertices.

INPUT:

• labels - default: True - if False, each edge is simply a pair (u,v) of vertices.
• sort - default: True - if True, edges are sorted according to the default ordering.
• key - default: None - a function takes an edge (a pair or a triple, according to the labels keyword) as its one argument and returns a value that can be used for comparisons in the sorting algorithm.

OUTPUT: A list of tuples. It is safe to change the returned list.

Warning

Since any object may be a vertex, there is no guarantee that any two vertices will be comparable, and thus no guarantee how two edges may compare. With default objects for vertices (all integers), or when all the vertices are of the same simple type, then there should not be a problem with how the vertices will be sorted. However, if you need to guarantee a total order for the sorting of the edges, use the key argument, as illustrated in the examples below.

EXAMPLES:

sage: graphs.DodecahedralGraph().edges()
[(0, 1, None), (0, 10, None), (0, 19, None), (1, 2, None), (1, 8, None), (2, 3, None), (2, 6, None), (3, 4, None), (3, 19, None), (4, 5, None), (4, 17, None), (5, 6, None), (5, 15, None), (6, 7, None), (7, 8, None), (7, 14, None), (8, 9, None), (9, 10, None), (9, 13, None), (10, 11, None), (11, 12, None), (11, 18, None), (12, 13, None), (12, 16, None), (13, 14, None), (14, 15, None), (15, 16, None), (16, 17, None), (17, 18, None), (18, 19, None)]

sage: graphs.DodecahedralGraph().edges(labels=False)
[(0, 1), (0, 10), (0, 19), (1, 2), (1, 8), (2, 3), (2, 6), (3, 4), (3, 19), (4, 5), (4, 17), (5, 6), (5, 15), (6, 7), (7, 8), (7, 14), (8, 9), (9, 10), (9, 13), (10, 11), (11, 12), (11, 18), (12, 13), (12, 16), (13, 14), (14, 15), (15, 16), (16, 17), (17, 18), (18, 19)]

sage: D = graphs.DodecahedralGraph().to_directed()
sage: D.edges()
[(0, 1, None), (0, 10, None), (0, 19, None), (1, 0, None), (1, 2, None), (1, 8, None), (2, 1, None), (2, 3, None), (2, 6, None), (3, 2, None), (3, 4, None), (3, 19, None), (4, 3, None), (4, 5, None), (4, 17, None), (5, 4, None), (5, 6, None), (5, 15, None), (6, 2, None), (6, 5, None), (6, 7, None), (7, 6, None), (7, 8, None), (7, 14, None), (8, 1, None), (8, 7, None), (8, 9, None), (9, 8, None), (9, 10, None), (9, 13, None), (10, 0, None), (10, 9, None), (10, 11, None), (11, 10, None), (11, 12, None), (11, 18, None), (12, 11, None), (12, 13, None), (12, 16, None), (13, 9, None), (13, 12, None), (13, 14, None), (14, 7, None), (14, 13, None), (14, 15, None), (15, 5, None), (15, 14, None), (15, 16, None), (16, 12, None), (16, 15, None), (16, 17, None), (17, 4, None), (17, 16, None), (17, 18, None), (18, 11, None), (18, 17, None), (18, 19, None), (19, 0, None), (19, 3, None), (19, 18, None)]
sage: D.edges(labels = False)
[(0, 1), (0, 10), (0, 19), (1, 0), (1, 2), (1, 8), (2, 1), (2, 3), (2, 6), (3, 2), (3, 4), (3, 19), (4, 3), (4, 5), (4, 17), (5, 4), (5, 6), (5, 15), (6, 2), (6, 5), (6, 7), (7, 6), (7, 8), (7, 14), (8, 1), (8, 7), (8, 9), (9, 8), (9, 10), (9, 13), (10, 0), (10, 9), (10, 11), (11, 10), (11, 12), (11, 18), (12, 11), (12, 13), (12, 16), (13, 9), (13, 12), (13, 14), (14, 7), (14, 13), (14, 15), (15, 5), (15, 14), (15, 16), (16, 12), (16, 15), (16, 17), (17, 4), (17, 16), (17, 18), (18, 11), (18, 17), (18, 19), (19, 0), (19, 3), (19, 18)]

The default is to sort the returned list in the default fashion, as in the above examples. this can be overridden by specifying a key function. This first example just ignores the labels in the third component of the triple.

sage: G=graphs.CycleGraph(5)
sage: G.edges(key = lambda x: (x[1],-x[0]))
[(0, 1, None), (1, 2, None), (2, 3, None), (3, 4, None), (0, 4, None)]

We set the labels to characters and then perform a default sort followed by a sort according to the labels.

sage: G=graphs.CycleGraph(5)
sage: for e in G.edges():
....:   G.set_edge_label(e[0], e[1], chr(ord('A')+e[0]+5*e[1]))
sage: G.edges(sort=True)
[(0, 1, 'F'), (0, 4, 'U'), (1, 2, 'L'), (2, 3, 'R'), (3, 4, 'X')]
sage: G.edges(key=lambda x: x[2])
[(0, 1, 'F'), (1, 2, 'L'), (2, 3, 'R'), (0, 4, 'U'), (3, 4, 'X')]

edges_incident(vertices=None, labels=True, sort=True)

Returns incident edges to some vertices.

If vertices` is a vertex, then it returns the list of edges incident to that vertex. If ``vertices is a list of vertices then it returns the list of all edges adjacent to those vertices. If vertices is None, returns a list of all edges in graph. For digraphs, only lists outward edges.

INPUT:

• vertices - object (default: None) - a vertex, a list of vertices or None.
• labels - bool (default: True) - if False, each edge is a tuple (u,v) of vertices.
• sort - bool (default: True) - if True the returned list is sorted.

EXAMPLES:

sage: graphs.PetersenGraph().edges_incident([0,9], labels=False)
[(0, 1), (0, 4), (0, 5), (4, 9), (6, 9), (7, 9)]
sage: D = DiGraph({0:[1]})
sage: D.edges_incident([0])
[(0, 1, None)]
sage: D.edges_incident([1])
[]

eigenspaces(laplacian=False)

Returns the right eigenspaces of the adjacency matrix of the graph.

INPUT:

• laplacian - if True, use the Laplacian matrix (see kirchhoff_matrix())

OUTPUT:

A list of pairs. Each pair is an eigenvalue of the adjacency matrix of the graph, followed by the vector space that is the eigenspace for that eigenvalue, when the eigenvectors are placed on the right of the matrix.

For some graphs, some of the eigenspaces are described exactly by vector spaces over a NumberField(). For numerical eigenvectors use eigenvectors().

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.eigenspaces()
[
(3, Vector space of degree 10 and dimension 1 over Rational Field
User basis matrix:
[1 1 1 1 1 1 1 1 1 1]),
(-2, Vector space of degree 10 and dimension 4 over Rational Field
User basis matrix:
[ 1  0  0  0 -1 -1 -1  0  1  1]
[ 0  1  0  0 -1  0 -2 -1  1  2]
[ 0  0  1  0 -1  1 -1 -2  0  2]
[ 0  0  0  1 -1  1  0 -1 -1  1]),
(1, Vector space of degree 10 and dimension 5 over Rational Field
User basis matrix:
[ 1  0  0  0  0  1 -1  0  0 -1]
[ 0  1  0  0  0 -1  1 -1  0  0]
[ 0  0  1  0  0  0 -1  1 -1  0]
[ 0  0  0  1  0  0  0 -1  1 -1]
[ 0  0  0  0  1 -1  0  0 -1  1])
]

Eigenspaces for the Laplacian should be identical since the Petersen graph is regular. However, since the output also contains the eigenvalues, the two outputs are slightly different.

sage: P.eigenspaces(laplacian=True)
[
(0, Vector space of degree 10 and dimension 1 over Rational Field
User basis matrix:
[1 1 1 1 1 1 1 1 1 1]),
(5, Vector space of degree 10 and dimension 4 over Rational Field
User basis matrix:
[ 1  0  0  0 -1 -1 -1  0  1  1]
[ 0  1  0  0 -1  0 -2 -1  1  2]
[ 0  0  1  0 -1  1 -1 -2  0  2]
[ 0  0  0  1 -1  1  0 -1 -1  1]),
(2, Vector space of degree 10 and dimension 5 over Rational Field
User basis matrix:
[ 1  0  0  0  0  1 -1  0  0 -1]
[ 0  1  0  0  0 -1  1 -1  0  0]
[ 0  0  1  0  0  0 -1  1 -1  0]
[ 0  0  0  1  0  0  0 -1  1 -1]
[ 0  0  0  0  1 -1  0  0 -1  1])
]

Notice how one eigenspace below is described with a square root of 2. For the two possible values (positive and negative) there is a corresponding eigenspace.

sage: C = graphs.CycleGraph(8)
sage: C.eigenspaces()
[
(2, Vector space of degree 8 and dimension 1 over Rational Field
User basis matrix:
[1 1 1 1 1 1 1 1]),
(-2, Vector space of degree 8 and dimension 1 over Rational Field
User basis matrix:
[ 1 -1  1 -1  1 -1  1 -1]),
(0, Vector space of degree 8 and dimension 2 over Rational Field
User basis matrix:
[ 1  0 -1  0  1  0 -1  0]
[ 0  1  0 -1  0  1  0 -1]),
(a3, Vector space of degree 8 and dimension 2 over Number Field in a3 with defining polynomial x^2 - 2
User basis matrix:
[  1   0  -1 -a3  -1   0   1  a3]
[  0   1  a3   1   0  -1 -a3  -1])
]

A digraph may have complex eigenvalues and eigenvectors. For a 3-cycle, we have:

sage: T = DiGraph({0:[1], 1:[2], 2:[0]})
sage: T.eigenspaces()
[
(1, Vector space of degree 3 and dimension 1 over Rational Field
User basis matrix:
[1 1 1]),
(a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomial x^2 + x + 1
User basis matrix:
[      1      a1 -a1 - 1])
]

eigenvectors(laplacian=False)

Returns the right eigenvectors of the adjacency matrix of the graph.

INPUT:

• laplacian - if True, use the Laplacian matrix (see kirchhoff_matrix())

OUTPUT:

A list of triples. Each triple begins with an eigenvalue of the adjacency matrix of the graph. This is followed by a list of eigenvectors for the eigenvalue, when the eigenvectors are placed on the right side of the matrix. Together, the eigenvectors form a basis for the eigenspace. The triple concludes with the algebraic multiplicity of the eigenvalue.

For some graphs, the exact eigenspaces provided by eigenspaces() provide additional insight into the structure of the eigenspaces.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.eigenvectors()
[(3, [
(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
], 1), (-2, [
(1, 0, 0, 0, -1, -1, -1, 0, 1, 1),
(0, 1, 0, 0, -1, 0, -2, -1, 1, 2),
(0, 0, 1, 0, -1, 1, -1, -2, 0, 2),
(0, 0, 0, 1, -1, 1, 0, -1, -1, 1)
], 4), (1, [
(1, 0, 0, 0, 0, 1, -1, 0, 0, -1),
(0, 1, 0, 0, 0, -1, 1, -1, 0, 0),
(0, 0, 1, 0, 0, 0, -1, 1, -1, 0),
(0, 0, 0, 1, 0, 0, 0, -1, 1, -1),
(0, 0, 0, 0, 1, -1, 0, 0, -1, 1)
], 5)]

Eigenspaces for the Laplacian should be identical since the Petersen graph is regular. However, since the output also contains the eigenvalues, the two outputs are slightly different.

sage: P.eigenvectors(laplacian=True)
[(0, [
(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
], 1), (5, [
(1, 0, 0, 0, -1, -1, -1, 0, 1, 1),
(0, 1, 0, 0, -1, 0, -2, -1, 1, 2),
(0, 0, 1, 0, -1, 1, -1, -2, 0, 2),
(0, 0, 0, 1, -1, 1, 0, -1, -1, 1)
], 4), (2, [
(1, 0, 0, 0, 0, 1, -1, 0, 0, -1),
(0, 1, 0, 0, 0, -1, 1, -1, 0, 0),
(0, 0, 1, 0, 0, 0, -1, 1, -1, 0),
(0, 0, 0, 1, 0, 0, 0, -1, 1, -1),
(0, 0, 0, 0, 1, -1, 0, 0, -1, 1)
], 5)]

sage: C = graphs.CycleGraph(8)
sage: C.eigenvectors()
[(2, [
(1, 1, 1, 1, 1, 1, 1, 1)
], 1), (-2, [
(1, -1, 1, -1, 1, -1, 1, -1)
], 1), (0, [
(1, 0, -1, 0, 1, 0, -1, 0),
(0, 1, 0, -1, 0, 1, 0, -1)
], 2), (-1.4142135623..., [(1, 0, -1, 1.4142135623..., -1, 0, 1, -1.4142135623...), (0, 1, -1.4142135623..., 1, 0, -1, 1.4142135623..., -1)], 2), (1.4142135623..., [(1, 0, -1, -1.4142135623..., -1, 0, 1, 1.4142135623...), (0, 1, 1.4142135623..., 1, 0, -1, -1.4142135623..., -1)], 2)]

A digraph may have complex eigenvalues. Previously, the complex parts of graph eigenvalues were being dropped. For a 3-cycle, we have:

sage: T = DiGraph({0:[1], 1:[2], 2:[0]})
sage: T.eigenvectors()
[(1, [
(1, 1, 1)
], 1), (-0.5000000000... - 0.8660254037...*I, [(1, -0.5000000000... - 0.8660254037...*I, -0.5000000000... + 0.8660254037...*I)], 1), (-0.5000000000... + 0.8660254037...*I, [(1, -0.5000000000... + 0.8660254037...*I, -0.5000000000... - 0.8660254037...*I)], 1)]

eulerian_circuit(return_vertices=False, labels=True, path=False)

Return a list of edges forming an Eulerian circuit if one exists. Otherwise return False.

This is implemented using Hierholzer’s algorithm.

INPUT:

• return_vertices – (default: False) optionally provide a list of vertices for the path
• labels – (default: True) whether to return edges with labels (3-tuples)
• path – (default: False) find an Eulerian path instead

OUTPUT:

either ([edges], [vertices]) or [edges] of an Eulerian circuit (or path)

EXAMPLES:

sage: g=graphs.CycleGraph(5);
sage: g.eulerian_circuit()
[(0, 4, None), (4, 3, None), (3, 2, None), (2, 1, None), (1, 0, None)]
sage: g.eulerian_circuit(labels=False)
[(0, 4), (4, 3), (3, 2), (2, 1), (1, 0)]

sage: g = graphs.CompleteGraph(7)
sage: edges, vertices = g.eulerian_circuit(return_vertices=True)
sage: vertices
[0, 6, 5, 4, 6, 3, 5, 2, 4, 3, 2, 6, 1, 5, 0, 4, 1, 3, 0, 2, 1, 0]

sage: graphs.CompleteGraph(4).eulerian_circuit()
False

A disconnected graph can be Eulerian:

sage: g = Graph({0: [], 1: [2], 2: [3], 3: [1], 4: []})
sage: g.eulerian_circuit(labels=False)
[(1, 3), (3, 2), (2, 1)]

sage: g = DiGraph({0: [1], 1: [2, 4], 2:[3], 3:[1]})
sage: g.eulerian_circuit(labels=False, path=True)
[(0, 1), (1, 2), (2, 3), (3, 1), (1, 4)]

sage: g = Graph({0:[1,2,3], 1:[2,3], 2:[3,4], 3:[4]})
sage: g.is_eulerian(path=True)
(0, 1)
sage: g.eulerian_circuit(labels=False, path=True)
[(1, 3), (3, 4), (4, 2), (2, 3), (3, 0), (0, 2), (2, 1), (1, 0)]

eulerian_orientation()

Returns a DiGraph which is an Eulerian orientation of the current graph.

An Eulerian graph being a graph such that any vertex has an even degree, an Eulerian orientation of a graph is an orientation of its edges such that each vertex \(v\) verifies \(d^+(v)=d^-(v)=d(v)/2\), where \(d^+\) and \(d^-\) respectively represent the out-degree and the in-degree of a vertex.

If the graph is not Eulerian, the orientation verifies for any vertex \(v\) that \(| d^+(v)-d^-(v) | \leq 1\).

ALGORITHM:

This algorithm is a random walk through the edges of the graph, which orients the edges according to the walk. When a vertex is reached which has no non-oriented edge ( this vertex must have odd degree ), the walk resumes at another vertex of odd degree, if any.

This algorithm has complexity \(O(m)\), where \(m\) is the number of edges in the graph.

EXAMPLES:

The CubeGraph with parameter 4, which is regular of even degree, has an Eulerian orientation such that \(d^+=d^-\):

sage: g=graphs.CubeGraph(4)
sage: g.degree()
[4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4]
sage: o=g.eulerian_orientation()
sage: o.in_degree()
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
sage: o.out_degree()
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

Secondly, the Petersen Graph, which is 3 regular has an orientation such that the difference between \(d^+\) and \(d^-\) is at most 1:

sage: g=graphs.PetersenGraph()
sage: o=g.eulerian_orientation()
sage: o.in_degree()
[2, 2, 2, 2, 2, 1, 1, 1, 1, 1]
sage: o.out_degree()
[1, 1, 1, 1, 1, 2, 2, 2, 2, 2]

export_to_file(filename, format=None, **kwds)

Export the graph to a file.

INPUT:

• filename (string) – a file name.

• format (string) – select the output format explicitly. If set to None (default), the format is set to be the file extension of filename. Admissible formats are: adjlist, dot, edgelist, gexf, gml, graphml, multiline_adjlist, pajek, yaml.

• All other arguments are forwarded to the subfunction. For more information, see their respective documentation:

  adjlist	http://networkx.lanl.gov/reference/generated/networkx.readwrite.adjlist.write_adjlist.html
  dot	https://networkx.github.io/documentation/latest/reference/generated/networkx.drawing.nx_pydot.write_dot.html
  edgelist	http://networkx.lanl.gov/reference/generated/networkx.readwrite.edgelist.write_edgelist.html
  gexf	http://networkx.lanl.gov/reference/generated/networkx.readwrite.gexf.write_gexf.html
  gml	http://networkx.lanl.gov/reference/generated/networkx.readwrite.gml.write_gml.html
  graphml	http://networkx.lanl.gov/reference/generated/networkx.readwrite.graphml.write_graphml.html
  multiline_adjlist	http://networkx.lanl.gov/reference/generated/networkx.readwrite.multiline_adjlist.write_multiline_adjlist.html
  pajek	http://networkx.lanl.gov/reference/generated/networkx.readwrite.pajek.write_pajek.html
  yaml	http://networkx.lanl.gov/reference/generated/networkx.readwrite.nx_yaml.write_yaml.html

See also

• save() – save a Sage object to a ‘sobj’ file (preserves all its attributes).

Note

This functions uses the write_* functions defined in NetworkX (see http://networkx.lanl.gov/reference/readwrite.html).

EXAMPLES:

sage: g = graphs.PetersenGraph()
sage: filename = tmp_filename(ext=".pajek")
sage: g.export_to_file(filename)
sage: import networkx
sage: G_networkx = networkx.read_pajek(filename)
sage: Graph(G_networkx).is_isomorphic(g)
True
sage: filename = tmp_filename(ext=".edgelist")
sage: g.export_to_file(filename, data=False)
sage: h = Graph(networkx.read_edgelist(filename))
sage: g.is_isomorphic(h)
True

faces(embedding=None)

Return the faces of an embedded graph.

A combinatorial embedding of a graph is a clockwise ordering of the neighbors of each vertex. From this information one can define the faces of the embedding, which is what this method returns.

INPUT:

• embedding - a combinatorial embedding dictionary. Format: {v1:[v2,v3], v2:[v1], v3:[v1]} (clockwise ordering of neighbors at each vertex). If set to None (default) the method will use the embedding stored as self._embedding. If none is stored, the method will compute the set of faces from the embedding returned by is_planar() (if the graph is, of course, planar).

Note

embedding is an ordered list based on the hash order of the vertices of graph. To avoid confusion, it might be best to set the rot_sys based on a ‘nice_copy’ of the graph.

See also

• set_embedding()
• get_embedding()
• is_planar()

EXAMPLES:

sage: T = graphs.TetrahedralGraph()
sage: T.faces({0: [1, 3, 2], 1: [0, 2, 3], 2: [0, 3, 1], 3: [0, 1, 2]})
[[(0, 1), (1, 2), (2, 0)],
 [(3, 2), (2, 1), (1, 3)],
 [(2, 3), (3, 0), (0, 2)],
 [(0, 3), (3, 1), (1, 0)]]

With no embedding provided:

sage: graphs.TetrahedralGraph().faces()
[[(0, 1), (1, 2), (2, 0)],
 [(3, 2), (2, 1), (1, 3)],
 [(2, 3), (3, 0), (0, 2)],
 [(0, 3), (3, 1), (1, 0)]]

With no embedding provided (non-planar graph):

sage: graphs.PetersenGraph().faces()
Traceback (most recent call last):
...
ValueError: No embedding is provided and the graph is not planar.

feedback_vertex_set(value_only=False, solver=None, verbose=0, constraint_generation=True)

Computes the minimum feedback vertex set of a (di)graph.

The minimum feedback vertex set of a (di)graph is a set of vertices that intersect all of its cycles. Equivalently, a minimum feedback vertex set of a (di)graph is a set \(S\) of vertices such that the digraph \(G-S\) is acyclic. For more information, see Wikipedia article Feedback_vertex_set.

INPUT:

• value_only – boolean (default: False)
  • When set to True, only the minimum cardinal of a minimum vertex set is returned.
  • When set to False, the Set of vertices of a minimal feedback vertex set is returned.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.
• constraint_generation (boolean) – whether to use constraint generation when solving the Mixed Integer Linear Program (default: True).

ALGORITHMS:

(Constraints generation)

When the parameter constraint_generation is enabled (default) the following MILP formulation is used to solve the problem:

\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_{v}\\ \mbox{Such that : }&\\ &\forall C\text{ circuits }\subseteq G, \sum_{v\in C}b_{v}\geq 1\\\end{split}\]

As the number of circuits contained in a graph is exponential, this LP is solved through constraint generation. This means that the solver is sequentially asked to solve the problem, knowing only a portion of the circuits contained in \(G\), each time adding to the list of its constraints the circuit which its last answer had left intact.

(Another formulation based on an ordering of the vertices)

When the graph is directed, a second (and very slow) formulation is available, which should only be used to check the result of the first implementation in case of doubt.

\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\\ &\forall (u,v)\in G, d_u-d_v+nb_u+nb_v\geq 0\\ &\forall u\in G, 0\leq d_u\leq |G|\\\end{split}\]

A brief explanation:

An acyclic digraph can be seen as a poset, and every poset has a linear extension. This means that in any acyclic digraph the vertices can be ordered with a total order \(<\) in such a way that if \((u,v)\in G\), then \(u<v\). Thus, this linear program is built in order to assign to each vertex \(v\) a number \(d_v\in [0,\dots,n-1]\) such that if there exists an edge \((u,v)\in G\) then either \(d_v<d_u\) or one of \(u\) or \(v\) is removed. The number of vertices removed is then minimized, which is the objective.

EXAMPLES:

The necessary example:

sage: g = graphs.PetersenGraph()
sage: fvs = g.feedback_vertex_set()
sage: len(fvs)
3
sage: g.delete_vertices(fvs)
sage: g.is_forest()
True

In a digraph built from a graph, any edge is replaced by arcs going in the two opposite directions, thus creating a cycle of length two. Hence, to remove all the cycles from the graph, each edge must see one of its neighbors removed: a feedback vertex set is in this situation a vertex cover:

sage: cycle = graphs.CycleGraph(5)
sage: dcycle = DiGraph(cycle)
sage: cycle.vertex_cover(value_only=True)
3
sage: feedback = dcycle.feedback_vertex_set()
sage: len(feedback)
3
sage: (u,v,l) = next(cycle.edge_iterator())
sage: u in feedback or v in feedback
True

For a circuit, the minimum feedback arc set is clearly \(1\):

sage: circuit = digraphs.Circuit(5)
sage: circuit.feedback_vertex_set(value_only=True) == 1
True

flow(x, y, value_only=True, integer=False, use_edge_labels=True, vertex_bound=False, algorithm=None, solver=None, verbose=0)

Returns a maximum flow in the graph from x to y represented by an optimal valuation of the edges. For more information, see the Wikipedia article on maximum flow.

As an optimization problem, is can be expressed this way :

\[\begin{split}\mbox{Maximize : }&\sum_{e\in G.edges()} w_e b_e\\ \mbox{Such that : }&\forall v \in G, \sum_{(u,v)\in G.edges()} b_{(u,v)}\leq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]

INPUT:

• x – Source vertex

• y – Sink vertex

• value_only – boolean (default: True)

  • When set to True, only the value of a maximal flow is returned.
  • When set to False, is returned a pair whose first element is the value of the maximum flow, and whose second value is a flow graph (a copy of the current graph, such that each edge has the flow using it as a label, the edges without flow being omitted).

• integer – boolean (default: True)

  • When set to True, computes an optimal solution under the constraint that the flow going through an edge has to be an integer.

• use_edge_labels – boolean (default: True)

  • When set to True, computes a maximum flow where each edge has a capacity defined by its label. (If an edge has no label, \(1\) is assumed.)
  • When set to False, each edge has capacity \(1\).

• vertex_bound – boolean (default: False)

  • When set to True, sets the maximum flow leaving a vertex different from \(x\) to \(1\) (useful for vertex connectivity parameters).

• algorithm – There are currently three different implementations of this method:

  • If algorithm = "FF", a Python implementation of the Ford-Fulkerson algorithm is used (only available when vertex_bound = False)
  • If algorithm = "LP", the flow problem is solved using Linear Programming.
  • If algorithm = "igraph", the igraph implementation of the Goldberg-Tarjan algorithm is used (only available when igraph is installed and vertex_bound = False)
  • If algorithm = None (default), we use LP if vertex_bound = True, otherwise, we use igraph if it is available, FF if it is not available.

• solver – Specify a Linear Program solver to be used. If set to None, the default one is used. function of MixedIntegerLinearProgram. See the documentation of MixedIntegerLinearProgram.solve for more information.

  Only useful when LP is used to solve the flow problem.

• verbose (integer) – sets the level of verbosity. Set to 0 by default (quiet).

  Only useful when LP is used to solve the flow problem.

Note

Even though the three different implementations are meant to return the same Flow values, they can not be expected to return the same Flow graphs.

Besides, the use of Linear Programming may possibly mean a (slight) numerical noise.

EXAMPLES:

Two basic applications of the flow method for the PappusGraph and the ButterflyGraph with parameter \(2\)

sage: g=graphs.PappusGraph()
sage: int(g.flow(1,2))
3

sage: b=digraphs.ButterflyGraph(2)
sage: int(b.flow(('00',1),('00',2)))
1

The flow method can be used to compute a matching in a bipartite graph by linking a source \(s\) to all the vertices of the first set and linking a sink \(t\) to all the vertices of the second set, then computing a maximum \(s-t\) flow

sage: g = DiGraph()
sage: g.add_edges([('s',i) for i in range(4)])
sage: g.add_edges([(i,4+j) for i in range(4) for j in range(4)])
sage: g.add_edges([(4+i,'t') for i in range(4)])
sage: [cardinal, flow_graph] = g.flow('s','t',integer=True,value_only=False)
sage: flow_graph.delete_vertices(['s','t'])
sage: len(flow_graph.edges())
4

The undirected case:

sage: g = Graph()
sage: g.add_edges([('s',i) for i in range(4)])
sage: g.add_edges([(i,4+j) for i in range(4) for j in range(4)])
sage: g.add_edges([(4+i,'t') for i in range(4)])
sage: [cardinal, flow_graph] = g.flow('s','t',integer=True,value_only=False)
sage: flow_graph.delete_vertices(['s','t'])
sage: len(flow_graph.edges())
4

genus(set_embedding=True, on_embedding=None, minimal=True, maximal=False, circular=None, ordered=True)

Returns the minimal genus of the graph.

The genus of a compact surface is the number of handles it has. The genus of a graph is the minimal genus of the surface it can be embedded into.

Note - This function uses Euler’s formula and thus it is necessary to consider only connected graphs.

INPUT:

• set_embedding (boolean) - whether or not to store an embedding attribute of the computed (minimal) genus of the graph. (Default is True).
• on_embedding (default: None) - two kinds of input are allowed:
  • a dictionary representing a combinatorial embedding on which the genus should be computed. Note that this must be a valid embedding for the graph. The dictionary structure is given by: vertex1: [neighbor1, neighbor2, neighbor3], vertex2: [neighbor] where there is a key for each vertex in the graph and a (clockwise) ordered list of each vertex’s neighbors as values. The value of on_embedding takes precedence over a stored _embedding attribute if minimal is set to False.
  • The value True, in order to indicate that the embedding stored as _embedding should be used (see examples).
• minimal (boolean) - whether or not to compute the minimal genus of the graph (i.e., testing all embeddings). If minimal is False, then either maximal must be True or on_embedding must not be None. If on_embedding is not None, it will take priority over minimal. Similarly, if maximal is True, it will take priority over minimal.
• maximal (boolean) - whether or not to compute the maximal genus of the graph (i.e., testing all embeddings). If maximal is False, then either minimal must be True or on_embedding must not be None. If on_embedding is not None, it will take priority over maximal. However, maximal takes priority over the default minimal.
• circular (list) - if circular is a list of vertices, the method computes the genus preserving a planar embedding of the this list. If circular is defined, on_embedding is not a valid option. It is set to None by default.
• ordered (boolean) - if circular is True, then whether or not the boundary order may be permuted. (Default is True, which means the boundary order is preserved.)

EXAMPLES:

sage: g = graphs.PetersenGraph()
sage: g.genus() # tests for minimal genus by default
1
sage: g.genus(on_embedding=True, maximal=True) # on_embedding overrides minimal and maximal arguments
1
sage: g.genus(maximal=True) # setting maximal to True overrides default minimal=True
3
sage: g.genus(on_embedding=g.get_embedding()) # can also send a valid combinatorial embedding dict
3
sage: (graphs.CubeGraph(3)).genus()
0
sage: K23 = graphs.CompleteBipartiteGraph(2,3)
sage: K23.genus()
0
sage: K33 = graphs.CompleteBipartiteGraph(3,3)
sage: K33.genus()
1

Using the circular argument, we can compute the minimal genus preserving a planar, ordered boundary:

sage: cube = graphs.CubeGraph(2)
sage: cube.genus(circular=['01','10'])
0
sage: cube.is_circular_planar()
True
sage: cube.genus(circular=['01','10'])
0
sage: cube.genus(circular=['01','10'], on_embedding=True)
0
sage: cube.genus(circular=['01','10'], maximal=True)
Traceback (most recent call last):
...
NotImplementedError: Cannot compute the maximal genus of a genus respecting a boundary.

Note: not everything works for multigraphs, looped graphs or digraphs. But the minimal genus is ultimately computable for every connected graph – but the embedding we obtain for the simple graph can’t be easily converted to an embedding of a non-simple graph. Also, the maximal genus of a multigraph does not trivially correspond to that of its simple graph.

sage: G = DiGraph({ 0 : [0,1,1,1], 1 : [2,2,3,3], 2 : [1,3,3], 3:[0,3]})
sage: G.genus()
Traceback (most recent call last):
...
NotImplementedError: Can't work with embeddings of non-simple graphs
sage: G.to_simple().genus()
0
sage: G.genus(set_embedding=False)
0
sage: G.genus(maximal=True, set_embedding=False)
Traceback (most recent call last):
...
NotImplementedError: Can't compute the maximal genus of a graph with loops or multiple edges

We break graphs with cut vertices into their blocks, which greatly speeds up computation of minimal genus. This is not implemented for maximal genus.

sage: K5 = graphs.CompleteGraph(5)
sage: G = K5.copy()
sage: s = 4
sage: for i in range(1,100):
....:     k = K5.relabel(list(range(s,s+5)),inplace=False)
....:     G.add_edges(k.edges())
....:     s += 4
sage: G.genus()
100

get_embedding()

Returns the attribute _embedding if it exists.

_embedding is a dictionary organized with vertex labels as keys and a list of each vertex’s neighbors in clockwise order.

Error-checked to insure valid embedding is returned.

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.genus()
1
sage: G.get_embedding()
{0: [1, 4, 5], 1: [0, 2, 6], 2: [1, 3, 7], 3: [2, 4, 8], 4: [0, 3, 9], 5: [0, 7, 8], 6: [1, 9, 8], 7: [2, 5, 9], 8: [3, 6, 5], 9: [4, 6, 7]}

get_pos(dim=2)

Returns the position dictionary, a dictionary specifying the coordinates of each vertex.

EXAMPLES: By default, the position of a graph is None:

sage: G = Graph()
sage: G.get_pos()
sage: G.get_pos() is None
True
sage: P = G.plot(save_pos=True)
sage: G.get_pos()
{}

Some of the named graphs come with a pre-specified positioning:

sage: G = graphs.PetersenGraph()
sage: G.get_pos()
{0: (...e-17, 1.0),
 ...
 9: (0.475..., 0.154...)}

get_vertex(vertex)

Retrieve the object associated with a given vertex.

INPUT:

• vertex - the given vertex

EXAMPLES:

sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() }
sage: d[2]
Moebius-Kantor Graph: Graph on 16 vertices
sage: T = graphs.TetrahedralGraph()
sage: T.vertices()
[0, 1, 2, 3]
sage: T.set_vertices(d)
sage: T.get_vertex(1)
Flower Snark: Graph on 20 vertices

get_vertices(verts=None)

Return a dictionary of the objects associated to each vertex.

INPUT:

• verts - iterable container of vertices

EXAMPLES:

sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() }
sage: T = graphs.TetrahedralGraph()
sage: T.set_vertices(d)
sage: T.get_vertices([1,2])
{1: Flower Snark: Graph on 20 vertices,
 2: Moebius-Kantor Graph: Graph on 16 vertices}

girth()

Computes the girth of the graph. For directed graphs, computes the girth of the undirected graph.

The girth is the length of the shortest cycle in the graph. Graphs without cycles have infinite girth.

EXAMPLES:

sage: graphs.TetrahedralGraph().girth()
3
sage: graphs.CubeGraph(3).girth()
4
sage: graphs.PetersenGraph().girth()
5
sage: graphs.HeawoodGraph().girth()
6
sage: next(graphs.trees(9)).girth()
+Infinity

See also

• odd_girth() – computes the odd girth of a graph.

graphplot(**options)

Returns a GraphPlot object.

EXAMPLES:

Creating a graphplot object uses the same options as graph.plot():

sage: g = Graph({}, loops=True, multiedges=True, sparse=True)
sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
....:     (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: GP = g.graphplot(edge_labels=True, color_by_label=True, edge_style='dashed')
sage: GP.plot()
Graphics object consisting of 26 graphics primitives

We can modify the graphplot object. Notice that the changes are cumulative:

sage: GP.set_edges(edge_style='solid')
sage: GP.plot()
Graphics object consisting of 26 graphics primitives
sage: GP.set_vertices(talk=True)
sage: GP.plot()
Graphics object consisting of 26 graphics primitives

graphviz_string(rankdir='down', edge_color=None, vertex_labels=True, edge_options=(), labels='string', color_by_label=False, edge_colors=None, edge_labels=False, subgraph_clusters=[], **options)

Return a representation in the \(dot\) language.

The \(dot\) language is a text based format for graphs. It is used by the software suite \(graphviz\). The specifications of the language are available on the web (see the reference [dotspec]).

INPUT:

• labels – “string” or “latex” (default: “string”). If labels is string latex command are not interpreted. This option stands for both vertex labels and edge labels.
• vertex_labels – boolean (default: True) whether to add the labels on vertices.
• edge_labels – boolean (default: False) whether to add the labels on edges.
• edge_color – (default: None) specify a default color for the edges.
• edge_colors – (default: None) a dictionary whose keys are colors and values are list of edges. The list of edges need not to be complete in which case the default color is used.
• color_by_label – (default: False) a boolean or dictionary or function whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible with edge_color and edge_colors.
• edge_options – a function (or tuple thereof) mapping edges to a dictionary of options for this edge.
• rankdir – ‘left’, ‘right’, ‘up’, or ‘down’ (default: ‘down’, for consistency with \(graphviz\)): the preferred ranking direction for acyclic layouts; see the \(rankdir\) option of \(graphviz\).
• subgraph_clusters – (default: []) a list of lists of vertices, From [dotspec]: “If supported, the layout engine will do the layout so that the nodes belonging to the cluster are drawn together, with the entire drawing of the cluster contained within a bounding rectangle. Note that, for good and bad, cluster subgraphs are not part of the DOT language, but solely a syntactic convention adhered to by certain of the layout engines.”

EXAMPLES:

sage: G = Graph({0:{1:None,2:None}, 1:{0:None,2:None}, 2:{0:None,1:None,3:'foo'}, 3:{2:'foo'}},sparse=True)
sage: print(G.graphviz_string(edge_labels=True))
graph {
  node_0  [label="0"];
  node_1  [label="1"];
  node_2  [label="2"];
  node_3  [label="3"];

  node_0 -- node_1;
  node_0 -- node_2;
  node_1 -- node_2;
  node_2 -- node_3 [label="foo"];
}

A variant, with the labels in latex, for post-processing with dot2tex:

sage: print(G.graphviz_string(edge_labels=True,labels = "latex"))
graph {
  node [shape="plaintext"];
  node_0  [label=" ", texlbl="$0$"];
  node_1  [label=" ", texlbl="$1$"];
  node_2  [label=" ", texlbl="$2$"];
  node_3  [label=" ", texlbl="$3$"];

  node_0 -- node_1;
  node_0 -- node_2;
  node_1 -- node_2;
  node_2 -- node_3 [label=" ", texlbl="$\text{\texttt{foo}}$"];
}

Same, with a digraph and a color for edges:

sage: G = DiGraph({0:{1:None,2:None}, 1:{2:None}, 2:{3:'foo'}, 3:{}} ,sparse=True)
sage: print(G.graphviz_string(edge_color="red"))
digraph {
  node_0  [label="0"];
  node_1  [label="1"];
  node_2  [label="2"];
  node_3  [label="3"];

edge [color="red"];
  node_0 -> node_1;
  node_0 -> node_2;
  node_1 -> node_2;
  node_2 -> node_3;
}

A digraph using latex labels for vertices and edges:

sage: f(x) = -1/x
sage: g(x) = 1/(x+1)
sage: G = DiGraph()
sage: G.add_edges([(i,f(i),f) for i in (1,2,1/2,1/4)])
sage: G.add_edges([(i,g(i),g) for i in (1,2,1/2,1/4)])
sage: print(G.graphviz_string(labels="latex",edge_labels=True))  # random
digraph {
  node [shape="plaintext"];
  node_10  [label=" ", texlbl="$1$"];
  node_11  [label=" ", texlbl="$2$"];
  node_3  [label=" ", texlbl="$-\frac{1}{2}$"];
  node_6  [label=" ", texlbl="$\frac{1}{2}$"];
  node_7  [label=" ", texlbl="$\frac{1}{2}$"];
  node_5  [label=" ", texlbl="$\frac{1}{3}$"];
  node_8  [label=" ", texlbl="$\frac{2}{3}$"];
  node_4  [label=" ", texlbl="$\frac{1}{4}$"];
  node_1  [label=" ", texlbl="$-2$"];
  node_9  [label=" ", texlbl="$\frac{4}{5}$"];
  node_0  [label=" ", texlbl="$-4$"];
  node_2  [label=" ", texlbl="$-1$"];

  node_10 -> node_2 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"];
  node_10 -> node_6 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"];
  node_11 -> node_3 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"];
  node_11 -> node_5 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"];
  node_7 -> node_1 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"];
  node_7 -> node_8 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"];
  node_4 -> node_0 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"];
  node_4 -> node_9 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"];
}

sage: print(G.graphviz_string(labels="latex",color_by_label=True))  # random
digraph {
  node [shape="plaintext"];
  node_10  [label=" ", texlbl="$1$"];
  node_11  [label=" ", texlbl="$2$"];
  node_3  [label=" ", texlbl="$-\frac{1}{2}$"];
  node_6  [label=" ", texlbl="$\frac{1}{2}$"];
  node_7  [label=" ", texlbl="$\frac{1}{2}$"];
  node_5  [label=" ", texlbl="$\frac{1}{3}$"];
  node_8  [label=" ", texlbl="$\frac{2}{3}$"];
  node_4  [label=" ", texlbl="$\frac{1}{4}$"];
  node_1  [label=" ", texlbl="$-2$"];
  node_9  [label=" ", texlbl="$\frac{4}{5}$"];
  node_0  [label=" ", texlbl="$-4$"];
  node_2  [label=" ", texlbl="$-1$"];

  node_10 -> node_2 [color = "#ff0000"];
  node_10 -> node_6 [color = "#00ffff"];
  node_11 -> node_3 [color = "#ff0000"];
  node_11 -> node_5 [color = "#00ffff"];
  node_7 -> node_1 [color = "#ff0000"];
  node_7 -> node_8 [color = "#00ffff"];
  node_4 -> node_0 [color = "#ff0000"];
  node_4 -> node_9 [color = "#00ffff"];
}

sage: print(G.graphviz_string(labels="latex",color_by_label={ f: "red", g: "blue" }))  # random
digraph {
  node [shape="plaintext"];
  node_10  [label=" ", texlbl="$1$"];
  node_11  [label=" ", texlbl="$2$"];
  node_3  [label=" ", texlbl="$-\frac{1}{2}$"];
  node_6  [label=" ", texlbl="$\frac{1}{2}$"];
  node_7  [label=" ", texlbl="$\frac{1}{2}$"];
  node_5  [label=" ", texlbl="$\frac{1}{3}$"];
  node_8  [label=" ", texlbl="$\frac{2}{3}$"];
  node_4  [label=" ", texlbl="$\frac{1}{4}$"];
  node_1  [label=" ", texlbl="$-2$"];
  node_9  [label=" ", texlbl="$\frac{4}{5}$"];
  node_0  [label=" ", texlbl="$-4$"];
  node_2  [label=" ", texlbl="$-1$"];

  node_10 -> node_2 [color = "red"];
  node_10 -> node_6 [color = "blue"];
  node_11 -> node_3 [color = "red"];
  node_11 -> node_5 [color = "blue"];
  node_7 -> node_1 [color = "red"];
  node_7 -> node_8 [color = "blue"];
  node_4 -> node_0 [color = "red"];
  node_4 -> node_9 [color = "blue"];
}

By default graphviz renders digraphs using a hierarchical layout, ranking the vertices down from top to bottom. Here we specify alternative ranking directions for this layout:

sage: D = DiGraph([[1,2]])
sage: print(D.graphviz_string(rankdir="up"))
digraph {
  rankdir=BT
  node_0  [label="1"];
  node_1  [label="2"];

  node_0 -> node_1;
}
sage: print(D.graphviz_string(rankdir="down"))
digraph {
  node_0  [label="1"];
  node_1  [label="2"];

  node_0 -> node_1;
}
sage: print(D.graphviz_string(rankdir="left"))
digraph {
  rankdir=RL
  node_0  [label="1"];
  node_1  [label="2"];

  node_0 -> node_1;
}
sage: print(D.graphviz_string(rankdir="right"))
digraph {
  rankdir=LR
  node_0  [label="1"];
  node_1  [label="2"];

  node_0 -> node_1;
}

Edge-specific options can also be specified by providing a function (or tuple thereof) which maps each edge to a dictionary of options. Valid options are “color”, “backward” (a boolean), “dot” (a string containing a sequence of options in dot format), “label” (a string), “label_style” (“string” or “latex”), “edge_string” (“–” or “->”). Here we state that the graph should be laid out so that edges starting from 1 are going backward (e.g. going up instead of down):

sage: def edge_options((u,v,label)):
....:     return { "backward": u == 1 }
sage: print(G.graphviz_string(edge_options = edge_options))  # random
digraph {
  node_10  [label="1"];
  node_11  [label="2"];
  node_3  [label="-1/2"];
  node_6  [label="1/2"];
  node_7  [label="1/2"];
  node_5  [label="1/3"];
  node_8  [label="2/3"];
  node_4  [label="1/4"];
  node_1  [label="-2"];
  node_9  [label="4/5"];
  node_0  [label="-4"];
  node_2  [label="-1"];

  node_2 -> node_10 [dir=back];
  node_6 -> node_10 [dir=back];
  node_11 -> node_3;
  node_11 -> node_5;
  node_7 -> node_1;
  node_7 -> node_8;
  node_4 -> node_0;
  node_4 -> node_9;
}

We now test all options:

sage: def edge_options((u,v,label)):
....:     options = { "color": { f: "red", g: "blue" }[label] }
....:     if (u,v) == (1/2, -2): options["label"]       = "coucou"; options["label_style"] = "string"
....:     if (u,v) == (1/2,2/3): options["dot"]         = "x=1,y=2"
....:     if (u,v) == (1,   -1): options["label_style"] = "latex"
....:     if (u,v) == (1,  1/2): options["edge_string"] = "<-"
....:     if (u,v) == (1/2,  1): options["backward"]    = True
....:     return options
sage: print(G.graphviz_string(edge_options = edge_options))  # random
digraph {
  node_10  [label="1"];
  node_11  [label="2"];
  node_3  [label="-1/2"];
  node_6  [label="1/2"];
  node_7  [label="1/2"];
  node_5  [label="1/3"];
  node_8  [label="2/3"];
  node_4  [label="1/4"];
  node_1  [label="-2"];
  node_9  [label="4/5"];
  node_0  [label="-4"];
  node_2  [label="-1"];

  node_10 -> node_2 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$", color = "red"];
  node_10 <- node_6 [color = "blue"];
  node_11 -> node_3 [color = "red"];
  node_11 -> node_5 [color = "blue"];
  node_7 -> node_1 [label="coucou", color = "red"];
  node_7 -> node_8 [x=1,y=2, color = "blue"];
  node_4 -> node_0 [color = "red"];
  node_4 -> node_9 [color = "blue"];
}

REFERENCES:

[dotspec]

(1, 2) http://www.graphviz.org/doc/info/lang.html

graphviz_to_file_named(filename, **options)

Write a representation in the dot in a file.

The dot language is a plaintext format for graph structures. See the documentation of graphviz_string() for available options.

INPUT:

filename - the name of the file to write in

options - options for the graphviz string

EXAMPLES:

sage: G = Graph({0:{1:None,2:None}, 1:{0:None,2:None}, 2:{0:None,1:None,3:'foo'}, 3:{2:'foo'}},sparse=True)
sage: tempfile = os.path.join(SAGE_TMP, 'temp_graphviz')
sage: G.graphviz_to_file_named(tempfile, edge_labels=True)
sage: print(open(tempfile).read())
graph {
  node_0  [label="0"];
  node_1  [label="1"];
  node_2  [label="2"];
  node_3  [label="3"];

  node_0 -- node_1;
  node_0 -- node_2;
  node_1 -- node_2;
  node_2 -- node_3 [label="foo"];
}

hamiltonian_cycle(algorithm='tsp')

Returns a Hamiltonian cycle/circuit of the current graph/digraph

A graph (resp. digraph) is said to be Hamiltonian if it contains as a subgraph a cycle (resp. a circuit) going through all the vertices.

Computing a Hamiltonian cycle/circuit being NP-Complete, this algorithm could run for some time depending on the instance.

ALGORITHM:

See Graph.traveling_salesman_problem for ‘tsp’ algorithm and find_hamiltonian from sage.graphs.generic_graph_pyx for ‘backtrack’ algorithm.

INPUT:

• algorithm - one of ‘tsp’ or ‘backtrack’.

OUTPUT:

If using the ‘tsp’ algorithm, returns a Hamiltonian cycle/circuit if it exists; otherwise, raises a EmptySetError exception. If using the ‘backtrack’ algorithm, returns a pair (B,P). If B is True then P is a Hamiltonian cycle and if B is False, P is a longest path found by the algorithm. Observe that if B is False, the graph may still be Hamiltonian. The ‘backtrack’ algorithm is only implemented for undirected graphs.

Warning

The ‘backtrack’ algorithm may loop endlessly on graphs with vertices of degree 1.

NOTE:

This function, as is_hamiltonian, computes a Hamiltonian cycle if it exists: the user should NOT test for Hamiltonicity using is_hamiltonian before calling this function, as it would result in computing it twice.

The backtrack algorithm is only implemented for undirected graphs.

EXAMPLES:

The Heawood Graph is known to be Hamiltonian

sage: g = graphs.HeawoodGraph()
sage: g.hamiltonian_cycle()
TSP from Heawood graph: Graph on 14 vertices

The Petersen Graph, though, is not

sage: g = graphs.PetersenGraph()
sage: g.hamiltonian_cycle()
Traceback (most recent call last):
...
EmptySetError: The given graph is not Hamiltonian

Now, using the backtrack algorithm in the Heawood graph

sage: G=graphs.HeawoodGraph()
sage: G.hamiltonian_cycle(algorithm='backtrack')
(True, [11, 10, 1, 2, 3, 4, 9, 8, 7, 6, 5, 0, 13, 12])

And now in the Petersen graph

sage: G=graphs.PetersenGraph()
sage: G.hamiltonian_cycle(algorithm='backtrack')
(False, [6, 8, 5, 0, 1, 2, 7, 9, 4, 3])

Finally, we test the algorithm in a cube graph, which is Hamiltonian

sage: G=graphs.CubeGraph(3)
sage: G.hamiltonian_cycle(algorithm='backtrack')
(True, ['010', '110', '100', '000', '001', '101', '111', '011'])

hamiltonian_path(s=None, t=None, use_edge_labels=False, algorithm='MILP', solver=None, verbose=0)

Return a Hamiltonian path of the current graph/digraph.

A path is Hamiltonian if it goes through all the vertices exactly once. Computing a Hamiltonian path being NP-Complete, this algorithm could run for some time depending on the instance.

When use_edge_labels == True, this method returns a minimum weight hamiltonian path.

See also

• longest_path()
• hamiltonian_cycle()

INPUT:

• s – vertex (optional); if specified, then forces the source of the path (the method then returns a Hamiltonian path starting at s)
• t – vertex (optional); if specified, then forces the destination of the path (the method then returns a Hamiltonian path ending at t)
• use_edge_labels – boolean (default: False); whether the labels on the edges are to be considered as weights (a label set to None or {} being considered as a weight of \(1\))
• algorithm – one of "MILP" (default) or "backtrack"; two remarks on this respect:
  • While the MILP formulation returns an exact answer, the backtrack algorithm is a randomized heuristic.
  • The backtrack algorithm does not support edge weighting, so setting use_edge_labels=True will force the use of the MILP algorithm.
• solver – (default: None) specify the Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve
• verbose – integer (default: 0); sets the level of verbosity with 0 meaning quiet

OUTPUT:

A subgraph of self corresponding to a (directed if self is directed) hamiltonian path. If no hamiltonian path is found, return None. If use_edge_labels == True, a pair weight, path is returned.

EXAMPLES:

The \(3 \times 3\)-grid has an Hamiltonian path, an hamiltonian path starting from vertex \((0, 0)\) and ending at vertex \((2, 2)\), but no Hamiltonian path starting from \((0, 0)\) and ending at \((0, 1)\):

sage: g = graphs.Grid2dGraph(3, 3)
sage: g.hamiltonian_path()
Subgraph of (2D Grid Graph for [3, 3]): Graph on 9 vertices
sage: g.hamiltonian_path(s=(0,0), t=(2,2))
Subgraph of (2D Grid Graph for [3, 3]): Graph on 9 vertices
sage: g.hamiltonian_path(s=(0,0), t=(2,2), use_edge_labels=True)
(8, Subgraph of (2D Grid Graph for [3, 3]): Graph on 9 vertices)
sage: g.hamiltonian_path(s=(0,0), t=(0,1)) is None
True
sage: g.hamiltonian_path(s=(0,0), t=(0,1), use_edge_labels=True)
(0, None)

has_edge(u, v=None, label=None)

Returns True if (u, v) is an edge, False otherwise.

INPUT: The following forms are accepted by NetworkX:

• G.has_edge( 1, 2 )
• G.has_edge( (1, 2) )
• G.has_edge( 1, 2, ‘label’ )
• G.has_edge( (1, 2, ‘label’) )

EXAMPLES:

sage: graphs.EmptyGraph().has_edge(9,2)
False
sage: DiGraph().has_edge(9,2)
False
sage: G = Graph(sparse=True)
sage: G.add_edge(0,1,"label")
sage: G.has_edge(0,1,"different label")
False
sage: G.has_edge(0,1,"label")
True

has_loops()

Return whether there are loops in the (di)graph

EXAMPLES:

sage: G = Graph(loops=True); G
Looped graph on 0 vertices
sage: G.has_loops()
False
sage: G.allows_loops()
True
sage: G.add_edge((0,0))
sage: G.has_loops()
True
sage: G.loops()
[(0, 0, None)]
sage: G.allow_loops(False); G
Graph on 1 vertex
sage: G.has_loops()
False
sage: G.edges()
[]

sage: D = DiGraph(loops=True); D
Looped digraph on 0 vertices
sage: D.has_loops()
False
sage: D.allows_loops()
True
sage: D.add_edge((0,0))
sage: D.has_loops()
True
sage: D.loops()
[(0, 0, None)]
sage: D.allow_loops(False); D
Digraph on 1 vertex
sage: D.has_loops()
False
sage: D.edges()
[]

has_multiple_edges(to_undirected=False)

Returns whether there are multiple edges in the (di)graph.

INPUT:

• to_undirected – (default: False) If True, runs the test on the undirected version of a DiGraph. Otherwise, treats DiGraph edges (u,v) and (v,u) as unique individual edges.

EXAMPLES:

sage: G = Graph(multiedges=True,sparse=True); G
Multi-graph on 0 vertices
sage: G.has_multiple_edges()
False
sage: G.allows_multiple_edges()
True
sage: G.add_edges([(0,1)]*3)
sage: G.has_multiple_edges()
True
sage: G.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: G.allow_multiple_edges(False); G
Graph on 2 vertices
sage: G.has_multiple_edges()
False
sage: G.edges()
[(0, 1, None)]

sage: D = DiGraph(multiedges=True,sparse=True); D
Multi-digraph on 0 vertices
sage: D.has_multiple_edges()
False
sage: D.allows_multiple_edges()
True
sage: D.add_edges([(0,1)]*3)
sage: D.has_multiple_edges()
True
sage: D.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: D.allow_multiple_edges(False); D
Digraph on 2 vertices
sage: D.has_multiple_edges()
False
sage: D.edges()
[(0, 1, None)]

sage: G = DiGraph({1:{2: 'h'}, 2:{1:'g'}},sparse=True)
sage: G.has_multiple_edges()
False
sage: G.has_multiple_edges(to_undirected=True)
True
sage: G.multiple_edges()
[]
sage: G.multiple_edges(to_undirected=True)
[(1, 2, 'h'), (2, 1, 'g')]

A loop is not a multiedge:

sage: g = Graph(loops=True,multiedges=True)
sage: g.add_edge(0,0)
sage: g.has_multiple_edges()
False

has_vertex(vertex)

Return True if vertex is one of the vertices of this graph.

INPUT:

• vertex - an integer

OUTPUT:

• bool - True or False

EXAMPLES:

sage: g = Graph({0:[1,2,3], 2:[4]}); g
Graph on 5 vertices
sage: 2 in g
True
sage: 10 in g
False
sage: graphs.PetersenGraph().has_vertex(99)
False

igraph_graph(vertex_attrs={}, edge_attrs={})

Converts the graph into an igraph graph.

Optionally, it is possible to add vertex attributes and edge attributes to the output graph.

Note

This routine needs the optional package igraph to be installed: to do so, it is enough to run sage -i python_igraph. For more information on the Python version of igraph, see http://igraph.org/python/.

INPUT:

• vertex_attrs (dictionary) - a dictionary where the key is a string (the attribute name), and the value is an iterable containing in position i the label of the ith vertex returned by vertices() (see http://igraph.org/python/doc/igraph.Graph-class.html#__init__ for more information).
• edge_attrs (dictionary) - a dictionary where the key is a string (the attribute name), and the value is an iterable containing in position i the label of the ith edge in the list outputted by edge_iterator() (see http://igraph.org/python/doc/igraph.Graph-class.html#__init__ for more information).

Note

In igraph, a graph is weighted if the edge labels have attribute weight. Hence, to create a weighted graph, it is enough to add this attribute.

Note

Often, Sage uses its own defined types for integer/floats. These types may not be igraph-compatible (see example below).

EXAMPLES:

Standard conversion:

sage: G = graphs.TetrahedralGraph() # optional - python_igraph
sage: H = G.igraph_graph()          # optional - python_igraph
sage: H.summary()                   # optional - python_igraph
'IGRAPH U--- 4 6 -- '
sage: G = digraphs.Path(3)          # optional - python_igraph
sage: H = G.igraph_graph()          # optional - python_igraph
sage: H.summary()                   # optional - python_igraph
'IGRAPH D--- 3 2 -- '

Adding edge attributes:

sage: G = Graph([(1,2,'a'),(2,3,'b')])                                     # optional - python_igraph
sage: H = G.igraph_graph(edge_attrs = {'label':[e[2] for e in G.edges()]}) # optional - python_igraph
sage: H.es['label']                                                        # optional - python_igraph
['a', 'b']

If edges have an attribute weight, the igraph graph is considered weighted:

sage: G = Graph([(1,2,{'weight':1}),(2,3,{'weight':2})])                              # optional - python_igraph
sage: H = G.igraph_graph(edge_attrs = {'weight':[e[2]['weight'] for e in G.edges()]}) # optional - python_igraph
sage: H.is_weighted()                                                                 # optional - python_igraph
True
sage: H.es['weight']                                                                  # optional - python_igraph
[1, 2]

Adding vertex attributes:

sage: G = graphs.GridGraph([2,2])                            # optional - python_igraph
sage: H = G.igraph_graph(vertex_attrs={'name':G.vertices()}) # optional - python_igraph
sage: H.vs()['name']                                         # optional - python_igraph
[(0, 0), (0, 1), (1, 0), (1, 1)]

Sometimes, Sage integer/floats are not compatible with igraph:

sage: G = Graph([(0,1,2)])                                                    # optional - python_igraph
sage: H = G.igraph_graph(edge_attrs = {'capacity':[e[2] for e in G.edges()]}) # optional - python_igraph
sage: H.maxflow_value(0, 1, 'capacity')                                       # optional - python_igraph
1.0
sage: H = G.igraph_graph(edge_attrs = {'capacity':[float(e[2]) for e in G.edges()]}) # optional - python_igraph
sage: H.maxflow_value(0, 1, 'capacity')                                              # optional - python_igraph
2.0

incidence_matrix(oriented=None, sparse=True)

Return the incidence matrix of the (di)graph.

Each row is a vertex, and each column is an edge. The vertices as ordered as obtained by the method vertices() and the edges as obtained by the method edge_iterator().

If the graph is not directed, then return a matrix with entries in \(\{0,1,2\}\). Each column will either contain two \(1\) (at the position of the endpoint of the edge), or one \(2\) (if the corresponding edge is a loop).

If the graph is directed return a matrix in \(\{-1,0,1\}\) where \(-1\) and \(+1\) correspond respectively to the source and the target of the edge. A loop will correspond to a zero column. In particular, it is not possible to recover the loops of an oriented graph from its incidence matrix.

See Wikipedia article Incidence_Matrix for more informations.

INPUT:

• oriented – an optional boolean. If set to True, the matrix will be oriented (i.e. with entries in \(-1\), \(0\), \(1\)) and if set to False the matrix will be not oriented (i.e. with entries in \(0\), \(1\), \(2\)). By default, this argument is inferred from the graph type. Note that in the case the graph is not directed and with the option directed=True, a somewhat random direction is chosen for each edge.
• sparse – default to True, whether to use a sparse or a dense matrix.

EXAMPLES:

sage: G = graphs.CubeGraph(3)
sage: G.incidence_matrix()
[0 1 0 0 0 1 0 1 0 0 0 0]
[0 0 0 1 0 1 1 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 0]
[1 0 0 1 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 1 0 0 1 0 1]
[0 0 1 0 0 0 0 0 1 0 1 0]
[0 0 0 0 1 0 0 0 1 1 0 0]
sage: G.incidence_matrix(oriented=True)
[ 0 -1  0  0  0 -1  0 -1  0  0  0  0]
[ 0  0  0 -1  0  1 -1  0  0  0  0  0]
[-1  1 -1  0  0  0  0  0  0  0  0  0]
[ 1  0  0  1 -1  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  1  0  0 -1 -1]
[ 0  0  0  0  0  0  1  0  0 -1  0  1]
[ 0  0  1  0  0  0  0  0 -1  0  1  0]
[ 0  0  0  0  1  0  0  0  1  1  0  0]

sage: G = digraphs.Circulant(4, [1,3])
sage: G.incidence_matrix()
[-1 -1  1  0  0  0  1  0]
[ 1  0 -1 -1  1  0  0  0]
[ 0  0  0  1 -1 -1  0  1]
[ 0  1  0  0  0  1 -1 -1]

sage: graphs.CompleteGraph(3).incidence_matrix()
[1 1 0]
[1 0 1]
[0 1 1]
sage: G = Graph([(0,0),(0,1),(0,1)], loops=True, multiedges=True)
sage: G.incidence_matrix(oriented=False)
[2 1 1]
[0 1 1]

A well known result states that the product of the (oriented) incidence matrix with its transpose of a (non-oriented graph) is in fact the Kirchhoff matrix:

sage: G = graphs.PetersenGraph()
sage: m = G.incidence_matrix(oriented=True)
sage: m * m.transpose() == G.kirchhoff_matrix()
True

sage: K = graphs.CompleteGraph(3)
sage: m = K.incidence_matrix(oriented=True)
sage: m * m.transpose() == K.kirchhoff_matrix()
True

sage: H = Graph([(0,0),(0,1),(0,1)], loops=True, multiedges=True)
sage: m = H.incidence_matrix(oriented=True)
sage: m * m.transpose() == H.kirchhoff_matrix()
True

is_cayley(return_group=False, mapping=False, generators=False, allow_disconnected=False)

Check whether the graph is a Cayley graph.

If none of the parameters are True, return a boolean indicating whether the graph is a Cayley graph. Otherwise, return a tuple containing said boolean and the requested data. If the graph is not a Cayley graph, each of the data will be None.

The empty graph is defined to be not a Cayley graph.

Note

For this routine to work on all graphs, the optional packages gap_packages and database_gap need to be installed: to do so, it is enough to run sage -i gap_packages database_gap.

INPUT:

• return_group (boolean; False) – If True, return a group for which the graph is a Cayley graph.
• mapping (boolean; False) – If True, return a mapping from vertices to group elements.
• generators (boolean; False) – If True, return the generating set of the Cayley graph.
• allow_disconnected (boolean; False) – If True, disconnected graphs are considered Cayley if they can be obtained from the Cayley construction with a generating set that does not generate the group.

ALGORITHM:

For connected graphs, find a regular subgroup of the automorphism group. For disconnected graphs, check that the graph is vertex-transitive and perform the check on one of its connected components. If a simple graph has density over 1/2, perform the check on its complement as its disconnectedness may increase performance.

EXAMPLES:

A Petersen Graph is not a Cayley graph:

sage: g = graphs.PetersenGraph()
sage: g.is_cayley()
False

A Cayley digraph is a Cayley graph:

sage: C7 = groups.permutation.Cyclic(7)
sage: S = [(1,2,3,4,5,6,7), (1,3,5,7,2,4,6), (1,5,2,6,3,7,4)]
sage: d = C7.cayley_graph(generators=S)
sage: d.is_cayley()
True

Graphs with loops and multiedges will have identity and repeated elements, respectively, among the generators:

sage: g = Graph(graphs.PaleyGraph(9), loops=True, multiedges=True)
sage: g.add_edges([(u, u) for u in g])
sage: g.add_edges([(u, u+1) for u in g])
sage: _, S = g.is_cayley(generators=True)
sage: S # random
[(),
 (0,2,1)(a,a + 2,a + 1)(2*a,2*a + 2,2*a + 1),
 (0,2,1)(a,a + 2,a + 1)(2*a,2*a + 2,2*a + 1),
 (0,1,2)(a,a + 1,a + 2)(2*a,2*a + 1,2*a + 2),
 (0,1,2)(a,a + 1,a + 2)(2*a,2*a + 1,2*a + 2),
 (0,2*a + 2,a + 1)(1,2*a,a + 2)(2,2*a + 1,a),
 (0,a + 1,2*a + 2)(1,a + 2,2*a)(2,a,2*a + 1)]

is_chordal(certificate=False, algorithm='B')

Tests whether the given graph is chordal.

A Graph \(G\) is said to be chordal if it contains no induced hole (a cycle of length at least 4).

Alternatively, chordality can be defined using a Perfect Elimination Order :

A Perfect Elimination Order of a graph \(G\) is an ordering \(v_1,...,v_n\) of its vertex set such that for all \(i\), the neighbors of \(v_i\) whose index is greater that \(i\) induce a complete subgraph in \(G\). Hence, the graph \(G\) can be totally erased by successively removing vertices whose neighborhood is a clique (also called simplicial vertices) [Fulkerson65].

(It can be seen that if \(G\) contains an induced hole, then it can not have a perfect elimination order. Indeed, if we write \(h_1,...,h_k\) the \(k\) vertices of such a hole, then the first of those vertices to be removed would have two non-adjacent neighbors in the graph.)

A Graph is then chordal if and only if it has a Perfect Elimination Order.

INPUT:

• certificate (boolean) – Whether to return a certificate.

  • If certificate = False (default), returns True or False accordingly.

  • If certificate = True, returns :

    • (True, peo) when the graph is chordal, where peo is a perfect elimination order of its vertices.
    • (False, Hole) when the graph is not chordal, where Hole (a Graph object) is an induced subgraph of self isomorphic to a hole.

• algorithm – Two algorithms are available for this method (see next section), which can be selected by setting algorithm = "A" or algorithm = "B" (default). While they will agree on whether the given graph is chordal, they can not be expected to return the same certificates.

ALGORITHM:

This algorithm works through computing a Lex BFS on the graph, then checking whether the order is a Perfect Elimination Order by computing for each vertex \(v\) the subgraph induces by its non-deleted neighbors, then testing whether this graph is complete.

This problem can be solved in \(O(m)\) [Rose75] ( where \(m\) is the number of edges in the graph ) but this implementation is not linear because of the complexity of Lex BFS.

Note

Because of a past bug (trac ticket #11735, trac ticket #11961), the first implementation (algorithm A) of this method sometimes returned as certificates subgraphs which were not holes. Since then, this bug has been fixed and the values are now double-checked before being returned, so that the algorithm only returns correct values or raises an exception. In the case where an exception is raised, the user is advised to switch to the other algorithm. And to please report the bug :-)

EXAMPLES:

The lexicographic product of a Path and a Complete Graph is chordal

sage: g = graphs.PathGraph(5).lexicographic_product(graphs.CompleteGraph(3))
sage: g.is_chordal()
True

The same goes with the product of a random lobster ( which is a tree ) and a Complete Graph

sage: g = graphs.RandomLobster(10,.5,.5).lexicographic_product(graphs.CompleteGraph(3))
sage: g.is_chordal()
True

The disjoint union of chordal graphs is still chordal:

sage: (2*g).is_chordal()
True

Let us check the certificate given by Sage is indeed a perfect elimination order:

sage: (_, peo) = g.is_chordal(certificate = True)
sage: for v in peo:
....:     if not g.subgraph(g.neighbors(v)).is_clique():
....:          print("This should never happen !")
....:     g.delete_vertex(v)

Of course, the Petersen Graph is not chordal as it has girth 5

sage: g = graphs.PetersenGraph()
sage: g.girth()
5
sage: g.is_chordal()
False

We can even obtain such a cycle as a certificate

sage: (_, hole) = g.is_chordal(certificate = True)
sage: hole
Subgraph of (Petersen graph): Graph on 5 vertices
sage: hole.is_isomorphic(graphs.CycleGraph(5))
True

REFERENCES:

[Rose75]

Rose, D.J. and Tarjan, R.E., Algorithmic aspects of vertex elimination, Proceedings of seventh annual ACM symposium on Theory of computing Page 254, ACM 1975

[Fulkerson65]

Fulkerson, D.R. and Gross, OA Incidence matrices and interval graphs Pacific J. Math 1965 Vol. 15, number 3, pages 835–855

is_circulant(certificate=False)

Tests whether the graph is circulant.

For more information on circulant graphs, see the Wikipedia page on circulant graphs.

INPUT:

• certificate (boolean) – whether to return a certificate for yes-answers. See OUTPUT section. Set to False by default.

OUTPUT:

When certificate is set to False (default) this method only returns True or False answers. When certificate is set to True, the method either returns (False, None) or (True, lists_of_parameters) each element of lists_of_parameters can be used to define the graph as a circulant graph.

See the documentation of CirculantGraph() and Circulant() for more information, and the examples below.

See also

CirculantGraph() – a constructor for circulant graphs.

EXAMPLES:

The Petersen graph is not a circulant graph:

sage: g = graphs.PetersenGraph()
sage: g.is_circulant()
False

A cycle is obviously a circulant graph, but several sets of parameters can be used to define it:

sage: g = graphs.CycleGraph(5)
sage: g.is_circulant(certificate = True)
(True, [(5, [1, 4]), (5, [2, 3])])

The same goes for directed graphs:

sage: g = digraphs.Circuit(5)
sage: g.is_circulant(certificate = True)
(True, [(5, [1]), (5, [3]), (5, [2]), (5, [4])])

With this information, it is very easy to create (and plot) all possible drawings of a circulant graph:

sage: g = graphs.CirculantGraph(13, [2, 3, 10, 11])
sage: for param in g.is_circulant(certificate = True)[1]:
....:    graphs.CirculantGraph(*param)
Circulant graph ([2, 3, 10, 11]): Graph on 13 vertices
Circulant graph ([1, 5, 8, 12]): Graph on 13 vertices
Circulant graph ([4, 6, 7, 9]): Graph on 13 vertices

is_circular_planar(on_embedding=None, kuratowski=False, set_embedding=True, boundary=None, ordered=False, set_pos=False)

Tests whether the graph is circular planar (outerplanar)

A graph is circular planar if it has a planar embedding in which all vertices can be drawn in order on a circle. This method can also be used to check the existence of a planar embedding in which the vertices of a specific set (the boundary) can be drawn on a circle, all other vertices being drawn inside of the circle. An order can be defined on the vertices of the boundary in order to define how they are to appear on the circle.

INPUT:

• kuratowski (boolean) - if set to True, returns a tuple with boolean first entry and the Kuratowski subgraph (i.e. an edge subdivision of \(K_5\) or \(K_{3,3}\)) as the second entry (see OUTPUT below). It is set to False by default.
• on_embedding (boolean) - the embedding dictionary to test planarity on. (i.e.: will return True or False only for the given embedding). It is set to False by default.
• set_embedding (boolean) - whether or not to set the instance field variable that contains a combinatorial embedding (clockwise ordering of neighbors at each vertex). This value will only be set if a circular planar embedding is found. It is stored as a Python dict: v1: [n1,n2,n3] where v1 is a vertex and n1,n2,n3 are its neighbors. It is set to True by default.
• boundary - a set of vertices that are required to be drawn on the circle, all others being drawn inside of it. It is set to None by default, meaning that all vertices should be drawn on the boundary.
• ordered (boolean) - whether or not to consider the order of the boundary. It is set to False by default, and required boundary to be defined.
• set_pos - whether or not to set the position dictionary (for plotting) to reflect the combinatorial embedding. Note that this value will default to False if set_emb is set to False. Also, the position dictionary will only be updated if a circular planar embedding is found.

OUTPUT:

The method returns True if the graph is circular planar, and False if it is not.

If kuratowski is set to True, then this function will return a tuple, whose first entry is a boolean and whose second entry is the Kuratowski subgraph (i.e. an edge subdivison of \(K_5\) or \(K_{3,3}\)) isolated by the Boyer-Myrvold algorithm. Note that this graph might contain a vertex or edges that were not in the initial graph. These would be elements referred to below as parts of the wheel and the star, which were added to the graph to require that the boundary can be drawn on the boundary of a disc, with all other vertices drawn inside (and no edge crossings).

ALGORITHM:

This is a linear time algorithm to test for circular planarity. It relies on the edge-addition planarity algorithm due to Boyer-Myrvold. We accomplish linear time for circular planarity by modifying the graph before running the general planarity algorithm.

REFERENCE:

[BM04]

John M. Boyer and Wendy J. Myrvold, On the Cutting Edge: Simplified O(n) Planarity by Edge Addition. Journal of Graph Algorithms and Applications, Vol. 8, No. 3, pp. 241-273, 2004.

EXAMPLES:

sage: g439 = Graph({1:[5,7], 2:[5,6], 3:[6,7], 4:[5,6,7]})
sage: g439.show()
sage: g439.is_circular_planar(boundary = [1,2,3,4])
False
sage: g439.is_circular_planar(kuratowski=True, boundary = [1,2,3,4])
(False, Graph on 8 vertices)
sage: g439.is_circular_planar(kuratowski=True, boundary = [1,2,3])
(True, None)
sage: g439.get_embedding()
{1: [7, 5],
2: [5, 6],
3: [6, 7],
4: [7, 6, 5],
5: [1, 4, 2],
6: [2, 4, 3],
7: [3, 4, 1]}

Order matters:

sage: K23 = graphs.CompleteBipartiteGraph(2,3)
sage: K23.is_circular_planar(boundary = [0,1,2,3])
True
sage: K23.is_circular_planar(ordered=True, boundary = [0,1,2,3])
False

With a different order:

sage: K23.is_circular_planar(set_embedding=True, boundary = [0,2,1,3])
True

is_clique(vertices=None, directed_clique=False)

Tests whether a set of vertices is a clique

A clique is a set of vertices such that there is an edge between any two vertices.

INPUT:

• vertices - Vertices can be a single vertex or an iterable container of vertices, e.g. a list, set, graph, file or numeric array. If not passed, defaults to the entire graph.
• directed_clique - (default False) If set to False, only consider the underlying undirected graph. If set to True and the graph is directed, only return True if all possible edges in _both_ directions exist.

EXAMPLES:

sage: g = graphs.CompleteGraph(4)
sage: g.is_clique([1,2,3])
True
sage: g.is_clique()
True
sage: h = graphs.CycleGraph(4)
sage: h.is_clique([1,2])
True
sage: h.is_clique([1,2,3])
False
sage: h.is_clique()
False
sage: i = graphs.CompleteGraph(4).to_directed()
sage: i.delete_edge([0,1])
sage: i.is_clique()
True
sage: i.is_clique(directed_clique=True)
False

is_connected()

Indicates whether the (di)graph is connected. Note that in a graph, path connected is equivalent to connected.

See also

• is_biconnected()

EXAMPLES:

sage: G = Graph( { 0 : [1, 2], 1 : [2], 3 : [4, 5], 4 : [5] } )
sage: G.is_connected()
False
sage: G.add_edge(0,3)
sage: G.is_connected()
True
sage: D = DiGraph( { 0 : [1, 2], 1 : [2], 3 : [4, 5], 4 : [5] } )
sage: D.is_connected()
False
sage: D.add_edge(0,3)
sage: D.is_connected()
True
sage: D = DiGraph({1:[0], 2:[0]})
sage: D.is_connected()
True

is_cut_edge(u, v=None, label=None)

Returns True if the input edge is a cut-edge or a bridge.

A cut edge (or bridge) is an edge that when removed increases the number of connected components. This function works with simple graphs as well as graphs with loops and multiedges. In a digraph, a cut edge is an edge that when removed increases the number of (weakly) connected components.

INPUT: The following forms are accepted

• G.is_cut_edge( 1, 2 )
• G.is_cut_edge( (1, 2) )
• G.is_cut_edge( 1, 2, ‘label’ )
• G.is_cut_edge( (1, 2, ‘label’) )

OUTPUT:

• Returns True if (u,v) is a cut edge, False otherwise

EXAMPLES:

sage: G = graphs.CompleteGraph(4)
sage: G.is_cut_edge(0,2)
False

sage: G = graphs.CompleteGraph(4)
sage: G.add_edge((0,5,'silly'))
sage: G.is_cut_edge((0,5,'silly'))
True

sage: G = Graph([[0,1],[0,2],[3,4],[4,5],[3,5]])
sage: G.is_cut_edge((0,1))
True

sage: G = Graph([[0,1],[0,2],[1,1]], loops = True)
sage: G.is_cut_edge((1,1))
False

sage: G = digraphs.Circuit(5)
sage: G.is_cut_edge((0,1))
False

sage: G = graphs.CompleteGraph(6)
sage: G.is_cut_edge((0,7))
Traceback (most recent call last):
...
ValueError: edge not in graph

is_cut_vertex(u, weak=False)

Returns True if the input vertex is a cut-vertex.

A vertex is a cut-vertex if its removal from the (di)graph increases the number of (strongly) connected components. Isolated vertices or leafs are not cut-vertices. This function works with simple graphs as well as graphs with loops and multiple edges.

INPUT:

• u – a vertex
• weak – (default: False) boolean set to \(True\) if the connectivity of directed graphs is to be taken in the weak sense, that is ignoring edges orientations.

OUTPUT:

Returns True if u is a cut-vertex, and False otherwise.

EXAMPLES:

Giving a LollipopGraph(4,2), that is a complete graph with 4 vertices with a pending edge:

sage: G = graphs.LollipopGraph(4,2)
sage: G.is_cut_vertex(0)
False
sage: G.is_cut_vertex(3)
True

Comparing the weak and strong connectivity of a digraph:

sage: D = digraphs.Circuit(6)
sage: D.is_strongly_connected()
True
sage: D.is_cut_vertex(2)
True
sage: D.is_cut_vertex(2, weak=True)
False

Giving a vertex that is not in the graph:

sage: G = graphs.CompleteGraph(6)
sage: G.is_cut_vertex(7)
Traceback (most recent call last):
...
ValueError: The input vertex is not in the vertex set.

is_cycle(directed_cycle=True)

Test whether self is a (directed) cycle graph.

We follow the definition provided in [BM2008] for undirected graphs. A cycle on three or more vertices is a simple graph whose vertices can be arranged in a cyclic order so that two vertices are adjacent if they are consecutive in the order, and not adjacent otherwise. A cycle on a vertex consists of a single vertex provided with a loop and a cycle with two vertices consists of two vertices connected by a pair of parallel edges. In other words, an undirected graph is a cycle if it is 2-regular and connected. The empty graph is not a cycle.

For directed graphs, a directed cycle, or circuit, on two or more vertices is a strongly connected directed graph without loops nor multiple edges with has many arcs as vertices. A circuit on a vertex consists of a single vertex provided with a loop.

INPUT:

• directed_cycle – (default True) If set to True and the graph is directed, only return True if self is a directed cycle graph (i.e., a circuit). If set to False, we ignore the direction of edges and so opposite arcs become multiple (parallel) edges. This parameter is ignored for undirected graphs.

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.is_cycle()
False
sage: graphs.CycleGraph(5).is_cycle()
True
sage: Graph([(0,1)]).is_cycle()
False
sage: Graph([(0,1), (0,1)], multiedges=True).is_cycle()
True
sage: Graph([(0,1), (0,1), (0,1)], multiedges=True).is_cycle()
False
sage: Graph().is_cycle()
False
sage: G = Graph(); G.allow_loops(True); G.add_edge(0,0)
sage: G.is_cycle()
True
sage: digraphs.Circuit(3).is_cycle()
True
sage: digraphs.Circuit(2).is_cycle()
True
sage: digraphs.Circuit(2).is_cycle(directed_cycle = False)
True
sage: D = DiGraph( graphs.CycleGraph(3) )
sage: D.is_cycle()
False
sage: D.is_cycle(directed_cycle = False)
False
sage: D.edges(labels = False)
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1)]

is_drawn_free_of_edge_crossings()

Returns True is the position dictionary for this graph is set and that position dictionary gives a planar embedding.

This simply checks all pairs of edges that don’t share a vertex to make sure that they don’t intersect.

Note

This function require that _pos attribute is set. (Returns False otherwise.)

EXAMPLES:

sage: D = graphs.DodecahedralGraph()
sage: D.set_planar_positions()
sage: D.is_drawn_free_of_edge_crossings()
True

is_equitable(partition, quotient_matrix=False)

Checks whether the given partition is equitable with respect to self.

A partition is equitable with respect to a graph if for every pair of cells C1, C2 of the partition, the number of edges from a vertex of C1 to C2 is the same, over all vertices in C1.

INPUT:

• partition - a list of lists
• quotient_matrix - (default False) if True, and the partition is equitable, returns a matrix over the integers whose rows and columns represent cells of the partition, and whose i,j entry is the number of vertices in cell j adjacent to each vertex in cell i (since the partition is equitable, this is well defined)

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8],[7]])
False
sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8,7]])
True
sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8,7]], quotient_matrix=True)
[1 2 0]
[1 0 2]
[0 2 1]

sage: ss = (graphs.WheelGraph(6)).line_graph(labels=False)
sage: prt = [[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]

sage: ss.is_equitable(prt)
Traceback (most recent call last):
...
TypeError: Partition ([[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]) is not valid for this graph: vertices are incorrect.

sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False)
sage: ss.is_equitable(prt)
False

is_eulerian(path=False)

Return true if the graph has a (closed) tour that visits each edge exactly once.

INPUT:

• path – by default this function finds if the graph contains a closed tour visiting each edge once, i.e. an Eulerian cycle. If you want to test the existence of an Eulerian path, set this argument to True. Graphs with this property are sometimes called semi-Eulerian.

OUTPUT:

True or False for the closed tour case. For an open tour search (path``=``True) the function returns False if the graph is not semi-Eulerian, or a tuple (u, v) in the other case. This tuple defines the edge that would make the graph Eulerian, i.e. close an existing open tour. This edge may or may not be already present in the graph.

EXAMPLES:

sage: graphs.CompleteGraph(4).is_eulerian()
False
sage: graphs.CycleGraph(4).is_eulerian()
True
sage: g = DiGraph({0:[1,2], 1:[2]}); g.is_eulerian()
False
sage: g = DiGraph({0:[2], 1:[3], 2:[0,1], 3:[2]}); g.is_eulerian()
True
sage: g = DiGraph({0:[1], 1:[2], 2:[0], 3:[]}); g.is_eulerian()
True
sage: g = Graph([(1,2), (2,3), (3,1), (4,5), (5,6), (6,4)]); g.is_eulerian()
False

sage: g = DiGraph({0: [1]}); g.is_eulerian(path=True)
(1, 0)
sage: graphs.CycleGraph(4).is_eulerian(path=True)
False
sage: g = DiGraph({0: [1], 1: [2,3], 2: [4]}); g.is_eulerian(path=True)
False

sage: g = Graph({0:[1,2,3], 1:[2,3], 2:[3,4], 3:[4]}, multiedges=True)
sage: g.is_eulerian()
False
sage: e = g.is_eulerian(path=True); e
(0, 1)
sage: g.add_edge(e)
sage: g.is_eulerian(path=False)
True
sage: g.is_eulerian(path=True)
False

is_gallai_tree()

Returns whether the current graph is a Gallai tree.

A graph is a Gallai tree if and only if it is connected and its \(2\)-connected components are all isomorphic to complete graphs or odd cycles.

A connected graph is not degree-choosable if and only if it is a Gallai tree [erdos1978choos].

REFERENCES:

[erdos1978choos]

Erdos, P. and Rubin, A.L. and Taylor, H. Proc. West Coast Conf. on Combinatorics Graph Theory and Computing, Congressus Numerantium vol 26, pages 125–157, 1979

EXAMPLES:

A complete graph is, or course, a Gallai Tree:

sage: g = graphs.CompleteGraph(15)
sage: g.is_gallai_tree()
True

The Petersen Graph is not:

sage: g = graphs.PetersenGraph()
sage: g.is_gallai_tree()
False

A Graph built from vertex-disjoint complete graphs linked by one edge to a special vertex \(-1\) is a ‘’star-shaped’’ Gallai tree

sage: g = 8 * graphs.CompleteGraph(6)
sage: g.add_edges([(-1,c[0]) for c in g.connected_components()])
sage: g.is_gallai_tree()
True

is_hamiltonian()

Tests whether the current graph is Hamiltonian.

A graph (resp. digraph) is said to be Hamiltonian if it contains as a subgraph a cycle (resp. a circuit) going through all the vertices.

Testing for Hamiltonicity being NP-Complete, this algorithm could run for some time depending on the instance.

ALGORITHM:

See Graph.traveling_salesman_problem.

OUTPUT:

Returns True if a Hamiltonian cycle/circuit exists, and False otherwise.

NOTE:

This function, as hamiltonian_cycle and traveling_salesman_problem, computes a Hamiltonian cycle if it exists: the user should NOT test for Hamiltonicity using is_hamiltonian before calling hamiltonian_cycle or traveling_salesman_problem as it would result in computing it twice.

EXAMPLES:

The Heawood Graph is known to be Hamiltonian

sage: g = graphs.HeawoodGraph()
sage: g.is_hamiltonian()
True

The Petergraph, though, is not

sage: g = graphs.PetersenGraph()
sage: g.is_hamiltonian()
False

is_immutable()

Returns whether the graph is immutable.

EXAMPLES:

sage: G = graphs.PetersenGraph() sage: G.is_immutable() False sage: Graph(G, immutable=True).is_immutable() True

is_independent_set(vertices=None)

Returns True if the set vertices is an independent set, False if not. An independent set is a set of vertices such that there is no edge between any two vertices.

INPUT:

• vertices - Vertices can be a single vertex or an iterable container of vertices, e.g. a list, set, graph, file or numeric array. If not passed, defaults to the entire graph.

EXAMPLES:

sage: graphs.CycleGraph(4).is_independent_set([1,3])
True
sage: graphs.CycleGraph(4).is_independent_set([1,2,3])
False

is_interval(certificate=False)

Check whether self is an interval graph

INPUT:

• certificate (boolean) – The function returns True or False according to the graph, when certificate = False (default). When certificate = True and the graph is an interval graph, a dictionary whose keys are the vertices and values are pairs of integers are returned instead of True. They correspond to an embedding of the interval graph, each vertex being represented by an interval going from the first of the two values to the second.

ALGORITHM:

Through the use of PQ-Trees

AUTHOR:

Nathann Cohen (implementation)

EXAMPLES:

A Petersen Graph is not chordal, nor can it be an interval graph

sage: g = graphs.PetersenGraph()
sage: g.is_interval()
False

Though we can build intervals from the corresponding random generator:

sage: g = graphs.RandomIntervalGraph(20)
sage: g.is_interval()
True

This method can also return, given an interval graph, a possible embedding (we can actually compute all of them through the PQ-Tree structures):

sage: g = Graph(':S__@[email protected][email protected][email protected][email protected][email protected][email protected]LMaCEGIJKNaCEGIJKNaCGIJKNPaCIP', loops=False, multiedges=False)
sage: d = g.is_interval(certificate = True)
sage: print(d)                                    # not tested
{0: (0, 20), 1: (1, 9), 2: (2, 36), 3: (3, 5), 4: (4, 38), 5: (6, 21), 6: (7, 27), 7: (8, 12), 8: (10, 29), 9: (11, 16), 10: (13, 39), 11: (14, 31), 12: (15, 32), 13: (17, 23), 14: (18, 22), 15: (19, 33), 16: (24, 25), 17: (26, 35), 18: (28, 30), 19: (34, 37)}

From this embedding, we can clearly build an interval graph isomorphic to the previous one:

sage: g2 = graphs.IntervalGraph(d.values())
sage: g2.is_isomorphic(g)
True

Enumerate all small interval graphs:

sage: n = 8
sage: count = [0]*(n+1)
sage: for g in graphs(n, augment='vertices',property= lambda x:x.is_interval()): # not tested -- 50s
....:     count[g.order()] += 1                                                  # not tested -- 50s
sage: count                                                                      # not tested -- 50s
[1, 1, 2, 4, 10, 27, 92, 369, 1807]

See also

• Interval Graph Recognition.
• PQ – Implementation of PQ-Trees.

is_isomorphic(other, certificate=False, verbosity=0, edge_labels=False)

Tests for isomorphism between self and other.

INPUT:

• certificate - if True, then output is \((a, b)\), where \(a\) is a boolean and \(b\) is either a map or None.
• edge_labels - default False, otherwise allows only permutations respecting edge labels.

OUTPUT:

• either a boolean or, if certificate is True, a tuple consisting of a boolean and a map or None

EXAMPLES:

Graphs:

sage: from sage.groups.perm_gps.permgroup_named import SymmetricGroup
sage: D = graphs.DodecahedralGraph()
sage: E = copy(D)
sage: gamma = SymmetricGroup(20).random_element()
sage: E.relabel(gamma)
sage: D.is_isomorphic(E)
True

sage: D = graphs.DodecahedralGraph()
sage: S = SymmetricGroup(20)
sage: gamma = S.random_element()
sage: E = copy(D)
sage: E.relabel(gamma)
sage: a,b = D.is_isomorphic(E, certificate=True); a
True
sage: from sage.plot.graphics import GraphicsArray
sage: from sage.graphs.generic_graph_pyx import spring_layout_fast
sage: position_D = spring_layout_fast(D)
sage: position_E = {}
sage: for vert in position_D:
....:  position_E[b[vert]] = position_D[vert]
sage: GraphicsArray([D.plot(pos=position_D), E.plot(pos=position_E)]).show() # long time

sage: g=graphs.HeawoodGraph()
sage: g.is_isomorphic(g)
True

Multigraphs:

sage: G = Graph(multiedges=True,sparse=True)
sage: G.add_edge((0,1,1))
sage: G.add_edge((0,1,2))
sage: G.add_edge((0,1,3))
sage: G.add_edge((0,1,4))
sage: H = Graph(multiedges=True,sparse=True)
sage: H.add_edge((3,4))
sage: H.add_edge((3,4))
sage: H.add_edge((3,4))
sage: H.add_edge((3,4))
sage: G.is_isomorphic(H)
True

Digraphs:

sage: A = DiGraph( { 0 : [1,2] } )
sage: B = DiGraph( { 1 : [0,2] } )
sage: A.is_isomorphic(B, certificate=True)
(True, {0: 1, 1: 0, 2: 2})

Edge labeled graphs:

sage: G = Graph(sparse=True)
sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] )
sage: H = G.relabel([1,2,3,4,0], inplace=False)
sage: G.is_isomorphic(H, edge_labels=True)
True

Edge labeled digraphs:

sage: G = DiGraph()
sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] )
sage: H = G.relabel([1,2,3,4,0], inplace=False)
sage: G.is_isomorphic(H, edge_labels=True)
True
sage: G.is_isomorphic(H, edge_labels=True, certificate=True)
(True, {0: 1, 1: 2, 2: 3, 3: 4, 4: 0})

is_planar(on_embedding=None, kuratowski=False, set_embedding=False, set_pos=False)

Test whether the graph is planar.

This wraps the reference implementation provided by John Boyer of the linear time planarity algorithm by edge addition due to Boyer Myrvold. (See reference code in planarity).

Note

The argument on_embedding takes precedence over set_embedding. This means that only the on_embedding combinatorial embedding will be tested for planarity and no _embedding attribute will be set as a result of this function call, unless on_embedding is None.

REFERENCE:

[1]	John M. Boyer and Wendy J. Myrvold, On the Cutting Edge: Simplified \(O(n)\) Planarity by Edge Addition. Journal of Graph Algorithms and Applications, Vol. 8, No. 3, pp. 241-273, 2004.

See also

• is_apex()

INPUT:

• kuratowski - returns a tuple with boolean as first entry. If the graph is nonplanar, will return the Kuratowski subgraph (i.e. an edge subdivision of \(K_5\) or \(K_{3,3}\)) as the second tuple entry. If the graph is planar, returns None as the second entry.
• on_embedding - the embedding dictionary to test planarity on. (i.e.: will return True or False only for the given embedding.)
• set_embedding - whether or not to set the instance field variable that contains a combinatorial embedding (clockwise ordering of neighbors at each vertex). This value will only be set if a planar embedding is found. It is stored as a Python dict: v1: [n1,n2,n3] where v1 is a vertex and n1,n2,n3 are its neighbors.
• set_pos - whether or not to set the position dictionary (for plotting) to reflect the combinatorial embedding. Note that this value will default to False if set_emb is set to False. Also, the position dictionary will only be updated if a planar embedding is found.

EXAMPLES:

sage: g = graphs.CubeGraph(4)
sage: g.is_planar()
False

sage: g = graphs.CircularLadderGraph(4)
sage: g.is_planar(set_embedding=True)
True
sage: g.get_embedding()
{0: [1, 4, 3],
 1: [2, 5, 0],
 2: [3, 6, 1],
 3: [0, 7, 2],
 4: [0, 5, 7],
 5: [1, 6, 4],
 6: [2, 7, 5],
 7: [4, 6, 3]}

sage: g = graphs.PetersenGraph()
sage: (g.is_planar(kuratowski=True))[1].adjacency_matrix()
[0 1 0 0 0 1 0 0 0]
[1 0 1 0 0 0 1 0 0]
[0 1 0 1 0 0 0 1 0]
[0 0 1 0 0 0 0 0 1]
[0 0 0 0 0 0 1 1 0]
[1 0 0 0 0 0 0 1 1]
[0 1 0 0 1 0 0 0 1]
[0 0 1 0 1 1 0 0 0]
[0 0 0 1 0 1 1 0 0]

sage: k43 = graphs.CompleteBipartiteGraph(4,3)
sage: result = k43.is_planar(kuratowski=True); result
(False, Graph on 6 vertices)
sage: result[1].is_isomorphic(graphs.CompleteBipartiteGraph(3,3))
True

Multi-edged and looped graphs are partially supported:

sage: G = Graph({0:[1,1]}, multiedges=True)
sage: G.is_planar()
True
sage: G.is_planar(on_embedding={})
Traceback (most recent call last):
...
NotImplementedError: Cannot compute with embeddings of multiple-edged or looped graphs.
sage: G.is_planar(set_pos=True)
Traceback (most recent call last):
...
NotImplementedError: Cannot compute with embeddings of multiple-edged or looped graphs.
sage: G.is_planar(set_embedding=True)
Traceback (most recent call last):
...
NotImplementedError: Cannot compute with embeddings of multiple-edged or looped graphs.
sage: G.is_planar(kuratowski=True)
(True, None)

sage: G = graphs.CompleteGraph(5)
sage: G = Graph(G, multiedges=True)
sage: G.add_edge(0,1)
sage: G.is_planar()
False
sage: b,k = G.is_planar(kuratowski=True)
sage: b
False
sage: k.vertices()
[0, 1, 2, 3, 4]

trac ticket #18045:

sage: g = graphs.CompleteGraph(4)
sage: g.is_planar(set_embedding=True)
True
sage: emb = {0 : [2,3,1], 1: [2,3,0], 2: [1,3,0], 3:[0,1,2]}
sage: g.is_planar(on_embedding=emb)
False

trac ticket #19193:

sage: Posets.BooleanLattice(3).cover_relations_graph().is_planar()
True

is_regular(k=None)

Return True if this graph is (\(k\)-)regular.

INPUT:

• k (default: None) - the degree of regularity to check for

EXAMPLES:

sage: G = graphs.HoffmanSingletonGraph()
sage: G.is_regular()
True
sage: G.is_regular(9)
False

So the Hoffman-Singleton graph is regular, but not 9-regular. In fact, we can now find the degree easily as follows:

sage: next(G.degree_iterator())
7

The house graph is not regular:

sage: graphs.HouseGraph().is_regular()
False

A graph without vertices is \(k\)-regular for every \(k\):

sage: Graph().is_regular()
True

is_subgraph(other, induced=True)

Return True if the graph is a subgraph of other, and False otherwise.

Warning

Please note that this method does not check whether self contains a subgraph isomorphic to other, but only if it directly contains it as a subgraph !

By default induced is True for backwards compatibility.

INPUT:

• induced - boolean (default: True) If set to True tests whether the graph is an induced subgraph of other that is if the vertices of the graph are also vertices of other, and the edges of the graph are equal to the edges of other between the vertices contained in the graph.

  If set to False tests whether the graph is a subgraph of other that is if all vertices of the graph are also in other and all edges of the graph are also in other.

OUTPUT:

boolean – True iff the graph is a (possibly induced) subgraph of other.

See also

If you are interested in the (possibly induced) subgraphs isomorphic to the graph in other, you are looking for the following methods:

• subgraph_search() – Find a subgraph isomorphic to other inside of the graph.
• subgraph_search_count() – Count the number of such copies.
• subgraph_search_iterator() – Iterate over all the copies of other contained in the graph.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: G = P.subgraph(list(range(6)))
sage: G.is_subgraph(P)
True

sage: H = graphs.CycleGraph(5)
sage: G = graphs.PathGraph(5)
sage: G.is_subgraph(H)
False
sage: G.is_subgraph(H, induced=False)
True
sage: H.is_subgraph(G, induced=False)
False

is_transitively_reduced()

Tests whether the digraph is transitively reduced.

A digraph is transitively reduced if it is equal to its transitive reduction.

EXAMPLES:

sage: d = DiGraph({0:[1],1:[2],2:[3]})
sage: d.is_transitively_reduced()
True

sage: d = DiGraph({0:[1,2],1:[2]})
sage: d.is_transitively_reduced()
False

sage: d = DiGraph({0:[1,2],1:[2],2:[]})
sage: d.is_transitively_reduced()
False

is_vertex_transitive(partition=None, verbosity=0, edge_labels=False, order=False, return_group=True, orbits=False)

Returns whether the automorphism group of self is transitive within the partition provided, by default the unit partition of the vertices of self (thus by default tests for vertex transitivity in the usual sense).

EXAMPLES:

sage: G = Graph({0:[1],1:[2]})
sage: G.is_vertex_transitive()
False
sage: P = graphs.PetersenGraph()
sage: P.is_vertex_transitive()
True
sage: D = graphs.DodecahedralGraph()
sage: D.is_vertex_transitive()
True
sage: R = graphs.RandomGNP(2000, .01)
sage: R.is_vertex_transitive()
False

kirchhoff_matrix(weighted=None, indegree=True, normalized=False, **kwds)

Returns the Kirchhoff matrix (a.k.a. the Laplacian) of the graph.

The Kirchhoff matrix is defined to be \(D - M\), where \(D\) is the diagonal degree matrix (each diagonal entry is the degree of the corresponding vertex), and \(M\) is the adjacency matrix. If normalized is True, then the returned matrix is \(D^{-1/2}(D-M)D^{-1/2}\).

( In the special case of DiGraphs, \(D\) is defined as the diagonal in-degree matrix or diagonal out-degree matrix according to the value of indegree)

INPUT:

• weighted – Binary variable :

  • If True, the weighted adjacency matrix is used for \(M\), and the diagonal matrix \(D\) takes into account the weight of edges (replace in the definition “degree” by “sum of the incident edges” ).
  • Else, each edge is assumed to have weight 1.

  Default is to take weights into consideration if and only if the graph is weighted.

• indegree – Binary variable :

  • If True, each diagonal entry of \(D\) is equal to the in-degree of the corresponding vertex.

  • Else, each diagonal entry of \(D\) is equal to the out-degree of the corresponding vertex.

    By default, indegree is set to True

  ( This variable only matters when the graph is a digraph )

• normalized – Binary variable :

  • If True, the returned matrix is \(D^{-1/2}(D-M)D^{-1/2}\), a normalized version of the Laplacian matrix. (More accurately, the normalizing matrix used is equal to \(D^{-1/2}\) only for non-isolated vertices. If vertex \(i\) is isolated, then diagonal entry \(i\) in the matrix is 1, rather than a division by zero.)
  • Else, the matrix \(D-M\) is returned

Note that any additional keywords will be passed on to either the adjacency_matrix or weighted_adjacency_matrix method.

AUTHORS:

• Tom Boothby
• Jason Grout

EXAMPLES:

sage: G = Graph(sparse=True)
sage: G.add_edges([(0,1,1),(1,2,2),(0,2,3),(0,3,4)])
sage: M = G.kirchhoff_matrix(weighted=True); M
[ 8 -1 -3 -4]
[-1  3 -2  0]
[-3 -2  5  0]
[-4  0  0  4]
sage: M = G.kirchhoff_matrix(); M
[ 3 -1 -1 -1]
[-1  2 -1  0]
[-1 -1  2  0]
[-1  0  0  1]
sage: M = G.laplacian_matrix(normalized=True); M
[                   1 -1/6*sqrt(3)*sqrt(2) -1/6*sqrt(3)*sqrt(2)         -1/3*sqrt(3)]
[-1/6*sqrt(3)*sqrt(2)                    1                 -1/2                    0]
[-1/6*sqrt(3)*sqrt(2)                 -1/2                    1                    0]
[        -1/3*sqrt(3)                    0                    0                    1]

sage: Graph({0:[],1:[2]}).laplacian_matrix(normalized=True)
[ 0  0  0]
[ 0  1 -1]
[ 0 -1  1]

A weighted directed graph with loops, changing the variable indegree

sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True)
sage: G.laplacian_matrix()
[ 4 -3]
[-4  3]

sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True)
sage: G.laplacian_matrix(indegree=False)
[ 3 -3]
[-4  4]

kronecker_product(other)

Returns the tensor product of self and other.

The tensor product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)\) equal to the Cartesian product of the vertices \(V(G)\) and \(V(H)\), and \(((u,v), (w,x))\) is an edge iff - \((u, w)\) is an edge of self, and - \((v, x)\) is an edge of other.

The tensor product is also known as the categorical product and the kronecker product (refering to the kronecker matrix product). See Wikipedia article on the Kronecker product.

EXAMPLES:

sage: Z = graphs.CompleteGraph(2)
sage: C = graphs.CycleGraph(5)
sage: T = C.tensor_product(Z); T
Graph on 10 vertices
sage: T.size()
10
sage: T.plot() # long time
Graphics object consisting of 21 graphics primitives

sage: D = graphs.DodecahedralGraph()
sage: P = graphs.PetersenGraph()
sage: T = D.tensor_product(P); T
Graph on 200 vertices
sage: T.size()
900
sage: T.plot() # long time
Graphics object consisting of 1101 graphics primitives

laplacian_matrix(weighted=None, indegree=True, normalized=False, **kwds)

Returns the Kirchhoff matrix (a.k.a. the Laplacian) of the graph.

The Kirchhoff matrix is defined to be \(D - M\), where \(D\) is the diagonal degree matrix (each diagonal entry is the degree of the corresponding vertex), and \(M\) is the adjacency matrix. If normalized is True, then the returned matrix is \(D^{-1/2}(D-M)D^{-1/2}\).

( In the special case of DiGraphs, \(D\) is defined as the diagonal in-degree matrix or diagonal out-degree matrix according to the value of indegree)

INPUT:

• weighted – Binary variable :

  • If True, the weighted adjacency matrix is used for \(M\), and the diagonal matrix \(D\) takes into account the weight of edges (replace in the definition “degree” by “sum of the incident edges” ).
  • Else, each edge is assumed to have weight 1.

  Default is to take weights into consideration if and only if the graph is weighted.

• indegree – Binary variable :

  • If True, each diagonal entry of \(D\) is equal to the in-degree of the corresponding vertex.

  • Else, each diagonal entry of \(D\) is equal to the out-degree of the corresponding vertex.

    By default, indegree is set to True

  ( This variable only matters when the graph is a digraph )

• normalized – Binary variable :

  • If True, the returned matrix is \(D^{-1/2}(D-M)D^{-1/2}\), a normalized version of the Laplacian matrix. (More accurately, the normalizing matrix used is equal to \(D^{-1/2}\) only for non-isolated vertices. If vertex \(i\) is isolated, then diagonal entry \(i\) in the matrix is 1, rather than a division by zero.)
  • Else, the matrix \(D-M\) is returned

Note that any additional keywords will be passed on to either the adjacency_matrix or weighted_adjacency_matrix method.

AUTHORS:

• Tom Boothby
• Jason Grout

EXAMPLES:

sage: G = Graph(sparse=True)
sage: G.add_edges([(0,1,1),(1,2,2),(0,2,3),(0,3,4)])
sage: M = G.kirchhoff_matrix(weighted=True); M
[ 8 -1 -3 -4]
[-1  3 -2  0]
[-3 -2  5  0]
[-4  0  0  4]
sage: M = G.kirchhoff_matrix(); M
[ 3 -1 -1 -1]
[-1  2 -1  0]
[-1 -1  2  0]
[-1  0  0  1]
sage: M = G.laplacian_matrix(normalized=True); M
[                   1 -1/6*sqrt(3)*sqrt(2) -1/6*sqrt(3)*sqrt(2)         -1/3*sqrt(3)]
[-1/6*sqrt(3)*sqrt(2)                    1                 -1/2                    0]
[-1/6*sqrt(3)*sqrt(2)                 -1/2                    1                    0]
[        -1/3*sqrt(3)                    0                    0                    1]

sage: Graph({0:[],1:[2]}).laplacian_matrix(normalized=True)
[ 0  0  0]
[ 0  1 -1]
[ 0 -1  1]

A weighted directed graph with loops, changing the variable indegree

sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True)
sage: G.laplacian_matrix()
[ 4 -3]
[-4  3]

sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True)
sage: G.laplacian_matrix(indegree=False)
[ 3 -3]
[-4  4]

latex_options()

Returns an instance of GraphLatex for the graph.

Changes to this object will affect the LaTeX version of the graph. For a full explanation of how to use LaTeX to render graphs, see the introduction to the graph_latex module.

EXAMPLES:

sage: g = graphs.PetersenGraph()
sage: opts = g.latex_options()
sage: opts
LaTeX options for Petersen graph: {}
sage: opts.set_option('tkz_style', 'Classic')
sage: opts
LaTeX options for Petersen graph: {'tkz_style': 'Classic'}

layout(layout=None, pos=None, dim=2, save_pos=False, **options)

Returns a layout for the vertices of this graph.

INPUT:

• layout – one of “acyclic”, “circular”, “ranked”, “graphviz”, “planar”, “spring”, or “tree”
• pos – a dictionary of positions or None (the default)
• save_pos – a boolean
• layout options – (see below)

If layout=algorithm is specified, this algorithm is used to compute the positions.

Otherwise, if pos is specified, use the given positions.

Otherwise, try to fetch previously computed and saved positions.

Otherwise use the default layout (usually the spring layout)

If save_pos = True, the layout is saved for later use.

EXAMPLES:

sage: g = digraphs.ButterflyGraph(1)
sage: g.layout()
{('0', 0): [2.22..., 0.832...],
 ('0', 1): [0.833..., 0.543...],
 ('1', 0): [1.12..., -0.830...],
 ('1', 1): [2.50..., -0.545...]}

sage: g.layout(layout="acyclic_dummy", save_pos=True)
{('0', 0): [0.3..., 0],
 ('0', 1): [0.3..., 1],
 ('1', 0): [0.6..., 0],
 ('1', 1): [0.6..., 1]}

sage: g.layout(dim = 3)
{('0', 0): [2.02..., 0.528..., 0.343...],
 ('0', 1): [1.61..., 0.260..., -0.927...],
 ('1', 0): [0.674..., -0.528..., -0.343...],
 ('1', 1): [1.07..., -0.260..., 0.927...]}

Here is the list of all the available layout options:

sage: from sage.graphs.graph_plot import layout_options
sage: for key, value in list(sorted(layout_options.items())):
....:     print("option {} : {}".format(key, value))
option by_component : Whether to do the spring layout by connected component -- a boolean.
option dim : The dimension of the layout -- 2 or 3.
option heights : A dictionary mapping heights to the list of vertices at this height.
option iterations : The number of times to execute the spring layout algorithm.
option layout : A layout algorithm -- one of : "acyclic", "circular" (plots the graph with vertices evenly distributed on a circle), "ranked", "graphviz", "planar", "spring" (traditional spring layout, using the graph's current positions as initial positions), or "tree" (the tree will be plotted in levels, depending on minimum distance for the root).
option prog : Which graphviz layout program to use -- one of "circo", "dot", "fdp", "neato", or "twopi".
option save_pos : Whether or not to save the computed position for the graph.
option spring : Use spring layout to finalize the current layout.
option tree_orientation : The direction of tree branches -- 'up', 'down', 'left' or 'right'.
option tree_root : A vertex designation for drawing trees. A vertex of the tree to be used as the root for the ``layout='tree'`` option. If no root is specified, then one is chosen close to the center of the tree. Ignored unless ``layout='tree'``

Some of them only apply to certain layout algorithms. For details, see layout_acyclic(), layout_planar(), layout_circular(), layout_spring(), ...

Warning

unknown optional arguments are silently ignored

Warning

graphviz and dot2tex are currently required to obtain a nice ‘acyclic’ layout. See layout_graphviz() for installation instructions.

A subclass may implement another layout algorithm \(blah\), by implementing a method .layout_blah. It may override the default layout by overriding layout_default(), and similarly override the predefined layouts.

Todo

use this feature for all the predefined graphs classes (like for the Petersen graph, ...), rather than systematically building the layout at construction time.

layout_circular(dim=2, **options)

Computes a circular layout for this graph

OUTPUT: a dictionary mapping vertices to positions

EXAMPLES:

sage: G = graphs.CirculantGraph(7,[1,3])
sage: G.layout_circular()
{0: [6.12...e-17, 1.0],
 1: [-0.78...,  0.62...],
 2: [-0.97..., -0.22...],
 3: [-0.43..., -0.90...],
 4: [0.43...,  -0.90...],
 5: [0.97...,  -0.22...],
 6: [0.78...,   0.62...]}
sage: G.plot(layout = "circular")
Graphics object consisting of 22 graphics primitives

layout_default(by_component=True, **options)

Computes a spring layout for this graph

INPUT:

• iterations – a positive integer
• dim – 2 or 3 (default: 2)

OUTPUT: a dictionary mapping vertices to positions

Returns a layout computed by randomly arranging the vertices along the given heights

EXAMPLES:

sage: g = graphs.LadderGraph(3) #TODO!!!!
sage: g.layout_spring()
{0: [1.28..., -0.94...],
 1: [1.57..., -0.10...],
 2: [1.83..., 0.74...],
 3: [0.53..., -0.75...],
 4: [0.79..., 0.10...],
 5: [1.08..., 0.94...]}
sage: g = graphs.LadderGraph(7)
sage: g.plot(layout = "spring")
Graphics object consisting of 34 graphics primitives

layout_extend_randomly(pos, dim=2)

Extends randomly a partial layout

INPUT:

• pos: a dictionary mapping vertices to positions

OUTPUT: a dictionary mapping vertices to positions

The vertices not referenced in pos are assigned random positions within the box delimited by the other vertices.

EXAMPLES:

sage: H = digraphs.ButterflyGraph(1)
sage: H.layout_extend_randomly({('0',0): (0,0), ('1',1): (1,1)})
{('0', 0): (0, 0),
 ('0', 1): [0.0446..., 0.332...],
 ('1', 0): [0.111..., 0.514...],
 ('1', 1): (1, 1)}

layout_graphviz(dim=2, prog='dot', **options)

Calls graphviz to compute a layout of the vertices of this graph.

INPUT:

• prog – one of “dot”, “neato”, “twopi”, “circo”, or “fdp”

EXAMPLES:

sage: g = digraphs.ButterflyGraph(2)
sage: g.layout_graphviz()  # optional - dot2tex graphviz
{('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...],
 ('...', ...): [...,...]}
sage: g.plot(layout = "graphviz")  # optional - dot2tex graphviz
Graphics object consisting of 29 graphics primitives

Note: the actual coordinates are not deterministic

By default, an acyclic layout is computed using graphviz‘s dot layout program. One may specify an alternative layout program:

sage: g.plot(layout = "graphviz", prog = "dot")   # optional - dot2tex graphviz
Graphics object consisting of 29 graphics primitives
sage: g.plot(layout = "graphviz", prog = "neato") # optional - dot2tex graphviz
Graphics object consisting of 29 graphics primitives
sage: g.plot(layout = "graphviz", prog = "twopi") # optional - dot2tex graphviz
Graphics object consisting of 29 graphics primitives
sage: g.plot(layout = "graphviz", prog = "fdp")   # optional - dot2tex graphviz
Graphics object consisting of 29 graphics primitives
sage: g = graphs.BalancedTree(5,2)
sage: g.plot(layout = "graphviz", prog = "circo")  # optional - dot2tex graphviz
Graphics object consisting of 62 graphics primitives

Todo

Put here some cool examples showcasing graphviz features.

This requires graphviz and the dot2tex spkg. Here are some installation tips:

• Install graphviz >= 2.14 so that the programs dot, neato, ... are in your path. The graphviz suite can be download from http://graphviz.org.
• Install dot2tex with sage -i dot2tex

Todo

Use the graphviz functionality of Networkx 1.0 once it will be merged into Sage.

layout_planar(set_embedding=False, on_embedding=None, external_face=None, test=False, circular=False, **options)

Uses Schnyder’s algorithm to compute a planar layout for self, raising an error if self is not planar.

INPUT:

• set_embedding - if True, sets the combinatorial embedding used (see self.get_embedding())
• on_embedding - dict: provide a combinatorial embedding
• external_face - ignored
• test - if True, perform sanity tests along the way
• circular - ignored

EXAMPLES:

sage: g = graphs.PathGraph(10)
sage: g.set_planar_positions(test=True)
True
sage: g = graphs.BalancedTree(3,4)
sage: g.set_planar_positions(test=True)
True
sage: g = graphs.CycleGraph(7)
sage: g.set_planar_positions(test=True)
True
sage: g = graphs.CompleteGraph(5)
sage: g.set_planar_positions(test=True,set_embedding=True)
Traceback (most recent call last):
...
ValueError: Complete graph is not a planar graph

layout_ranked(heights=None, dim=2, spring=False, **options)

Computes a ranked layout for this graph

INPUT:

• heights – a dictionary mapping heights to the list of vertices at this height

OUTPUT: a dictionary mapping vertices to positions

Returns a layout computed by randomly arranging the vertices along the given heights

EXAMPLES:

sage: g = graphs.LadderGraph(3)
sage: g.layout_ranked(heights = dict( (i,[i, i+3]) for i in range(3) ))
{0: [0.668..., 0],
 1: [0.667..., 1],
 2: [0.677..., 2],
 3: [1.34..., 0],
 4: [1.33..., 1],
 5: [1.33..., 2]}
sage: g = graphs.LadderGraph(7)
sage: g.plot(layout = "ranked", heights = dict( (i,[i, i+7]) for i in range(7) ))
Graphics object consisting of 34 graphics primitives

layout_spring(by_component=True, **options)

Computes a spring layout for this graph

INPUT:

• iterations – a positive integer
• dim – 2 or 3 (default: 2)

OUTPUT: a dictionary mapping vertices to positions

Returns a layout computed by randomly arranging the vertices along the given heights

EXAMPLES:

sage: g = graphs.LadderGraph(3) #TODO!!!!
sage: g.layout_spring()
{0: [1.28..., -0.94...],
 1: [1.57..., -0.10...],
 2: [1.83..., 0.74...],
 3: [0.53..., -0.75...],
 4: [0.79..., 0.10...],
 5: [1.08..., 0.94...]}
sage: g = graphs.LadderGraph(7)
sage: g.plot(layout = "spring")
Graphics object consisting of 34 graphics primitives

layout_tree(tree_orientation='down', tree_root=None, dim=2, **options)

Compute an ordered tree layout for this graph, which should be a tree (no non-oriented cycles).

INPUT:

• tree_root – the root vertex. By default None. In this case, a vertex is chosen close to the center of the tree.
• tree_orientation – the direction in which the tree is growing, can be 'up', 'down', 'left' or 'right' (default is 'down')

If the tree has been given a planar embedding (fixed circular order on the set of neighbors of every vertex) using set_embedding, the algorithm will create a layout that respects this embedding.

OUTPUT: a dictionary mapping vertices to positions

EXAMPLES:

sage: T = graphs.RandomLobster(25, 0.3, 0.3)
sage: T.show(layout='tree', tree_orientation='up')

sage: G = graphs.HoffmanSingletonGraph()
sage: T = Graph()
sage: T.add_edges(G.min_spanning_tree(starting_vertex=0))
sage: T.show(layout='tree', tree_root=0)

sage: G = graphs.BalancedTree(2, 2)
sage: G.layout_tree(tree_root=0)
{0: (1.5, 0),
 1: (2.5, -1),
 2: (0.5, -1),
 3: (3.0, -2),
 4: (2.0, -2),
 5: (1.0, -2),
 6: (0.0, -2)}

sage: G = graphs.BalancedTree(2,4)
sage: G.plot(layout="tree", tree_root = 0, tree_orientation = "up")
Graphics object consisting of 62 graphics primitives

sage: G = graphs.RandomTree(80)
sage: G.plot(layout="tree", tree_orientation = "right")
Graphics object consisting of 160 graphics primitives

Using the embedding when it exists:

sage: T = Graph([[0,1],[0,6],[0,3],[1,2],[1,5],[3,4],[3,7],[3,8]])
sage: T.set_embedding({0:[1,6,3],1:[2,5,0],2:[1],3:[4,7,8,0],
....:     4:[3],5:[1],6:[0],7:[3],8:[3]})
sage: T.layout_tree()
{0: (2.166..., 0),
 1: (3.5, -1),
 2: (4.0, -2),
 3: (1.0, -1),
 4: (2.0, -2),
 5: (3.0, -2),
 6: (2.0, -1),
 7: (1.0, -2),
 8: (0.0, -2)}
sage: T.plot(layout="tree", tree_root=3)
Graphics object consisting of 18 graphics primitives

lex_BFS(reverse=False, tree=False, initial_vertex=None)

Performs a Lex BFS on the graph.

A Lex BFS ( or Lexicographic Breadth-First Search ) is a Breadth First Search used for the recognition of Chordal Graphs. For more information, see the Wikipedia article on Lex-BFS.

INPUT:

• reverse (boolean) – whether to return the vertices in discovery order, or the reverse.

  False by default.

• tree (boolean) – whether to return the discovery directed tree (each vertex being linked to the one that saw it for the first time)

  False by default.

• initial_vertex – the first vertex to consider.

  None by default.

ALGORITHM:

This algorithm maintains for each vertex left in the graph a code corresponding to the vertices already removed. The vertex of maximal code ( according to the lexicographic order ) is then removed, and the codes are updated.

This algorithm runs in time \(O(n^2)\) ( where \(n\) is the number of vertices in the graph ), which is not optimal. An optimal algorithm would run in time \(O(m)\) ( where \(m\) is the number of edges in the graph ), and require the use of a doubly-linked list which are not available in python and can not really be written efficiently. This could be done in Cython, though.

EXAMPLES:

A Lex BFS is obviously an ordering of the vertices:

sage: g = graphs.PetersenGraph()
sage: len(g.lex_BFS()) == g.order()
True

For a Chordal Graph, a reversed Lex BFS is a Perfect Elimination Order

sage: g = graphs.PathGraph(3).lexicographic_product(graphs.CompleteGraph(2))
sage: g.lex_BFS(reverse=True)
[(2, 1), (2, 0), (1, 1), (1, 0), (0, 1), (0, 0)]

And the vertices at the end of the tree of discovery are, for chordal graphs, simplicial vertices (their neighborhood is a complete graph):

sage: g = graphs.ClawGraph().lexicographic_product(graphs.CompleteGraph(2))
sage: v = g.lex_BFS()[-1]
sage: peo, tree = g.lex_BFS(initial_vertex = v,  tree=True)
sage: leaves = [v for v in tree if tree.in_degree(v) ==0]
sage: all([g.subgraph(g.neighbors(v)).is_clique() for v in leaves])
True

lexicographic_product(other)

Returns the lexicographic product of self and other.

The lexicographic product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)=V(G)\times V(H)\), and \(((u,v), (w,x))\) is an edge iff :

• \((u, w)\) is an edge of \(G\), or
• \(u = w\) and \((v, x)\) is an edge of \(H\).

EXAMPLES:

sage: Z = graphs.CompleteGraph(2)
sage: C = graphs.CycleGraph(5)
sage: L = C.lexicographic_product(Z); L
Graph on 10 vertices
sage: L.plot() # long time
Graphics object consisting of 36 graphics primitives

sage: D = graphs.DodecahedralGraph()
sage: P = graphs.PetersenGraph()
sage: L = D.lexicographic_product(P); L
Graph on 200 vertices
sage: L.plot() # long time
Graphics object consisting of 3501 graphics primitives

line_graph(labels=True)

Returns the line graph of the (di)graph.

INPUT:

• labels (boolean) – whether edge labels should be taken in consideration. If labels=True, the vertices of the line graph will be triples (u,v,label), and pairs of vertices otherwise.

  This is set to True by default.

The line graph of an undirected graph G is an undirected graph H such that the vertices of H are the edges of G and two vertices e and f of H are adjacent if e and f share a common vertex in G. In other words, an edge in H represents a path of length 2 in G.

The line graph of a directed graph G is a directed graph H such that the vertices of H are the edges of G and two vertices e and f of H are adjacent if e and f share a common vertex in G and the terminal vertex of e is the initial vertex of f. In other words, an edge in H represents a (directed) path of length 2 in G.

Note

As a Graph object only accepts hashable objects as vertices (and as the vertices of the line graph are the edges of the graph), this code will fail if edge labels are not hashable. You can also set the argument labels=False to ignore labels.

See also

• The line_graph module.
• line_graph_forbidden_subgraphs() – the forbidden subgraphs of a line graph.
• is_line_graph() – tests whether a graph is a line graph.

EXAMPLES:

sage: g = graphs.CompleteGraph(4)
sage: h = g.line_graph()
sage: h.vertices()
[(0, 1, None),
(0, 2, None),
(0, 3, None),
(1, 2, None),
(1, 3, None),
(2, 3, None)]
sage: h.am()
[0 1 1 1 1 0]
[1 0 1 1 0 1]
[1 1 0 0 1 1]
[1 1 0 0 1 1]
[1 0 1 1 0 1]
[0 1 1 1 1 0]
sage: h2 = g.line_graph(labels=False)
sage: h2.vertices()
[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
sage: h2.am() == h.am()
True
sage: g = DiGraph([[1..4],lambda i,j: i<j])
sage: h = g.line_graph()
sage: h.vertices()
[(1, 2, None),
(1, 3, None),
(1, 4, None),
(2, 3, None),
(2, 4, None),
(3, 4, None)]
sage: h.edges()
[((1, 2, None), (2, 3, None), None),
 ((1, 2, None), (2, 4, None), None),
 ((1, 3, None), (3, 4, None), None),
 ((2, 3, None), (3, 4, None), None)]

Tests:

trac ticket #13787:

sage: g = graphs.KneserGraph(7,1)
sage: C = graphs.CompleteGraph(7)
sage: C.is_isomorphic(g)
True
sage: C.line_graph().is_isomorphic(g.line_graph())
True

longest_path(s=None, t=None, use_edge_labels=False, algorithm='MILP', solver=None, verbose=0)

Returns a longest path of self.

INPUT:

• s (vertex) – forces the source of the path (the method then returns the longest path starting at s). The argument is set to None by default, which means that no constraint is set upon the first vertex in the path.

• t (vertex) – forces the destination of the path (the method then returns the longest path ending at t). The argument is set to None by default, which means that no constraint is set upon the last vertex in the path.

• use_edge_labels (boolean) – whether the labels on the edges are to be considered as weights (a label set to None or {} being considered as a weight of \(1\)). Set to False by default.

• algorithm – one of "MILP" (default) or "backtrack". Two remarks on this respect:

  • While the MILP formulation returns an exact answer, the backtrack algorithm is a randomized heuristic.
  • As the backtrack algorithm does not support edge weighting, setting use_edge_labels=True will force the use of the MILP algorithm.

• solver – (default: None) Specify the Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

Note

The length of a path is assumed to be the number of its edges, or the sum of their labels.

OUTPUT:

A subgraph of self corresponding to a (directed if self is directed) longest path. If use_edge_labels == True, a pair weight, path is returned.

ALGORITHM:

Mixed Integer Linear Programming. (This problem is known to be NP-Hard).

EXAMPLES:

Petersen’s graph being hypohamiltonian, it has a longest path of length \(n-2\):

sage: g = graphs.PetersenGraph()
sage: lp = g.longest_path()
sage: lp.order() >= g.order() - 2
True

The heuristic totally agrees:

sage: g = graphs.PetersenGraph()
sage: g.longest_path(algorithm="backtrack").edges()
[(0, 1, None), (1, 2, None), (2, 3, None), (3, 4, None), (4, 9, None), (5, 7, None), (5, 8, None), (6, 8, None), (6, 9, None)]

Let us compute longest paths on random graphs with random weights. Each time, we ensure the resulting graph is indeed a path:

sage: for i in range(20):
....:     g = graphs.RandomGNP(15, 0.3)
....:     for u, v in g.edges(labels=False):
....:         g.set_edge_label(u, v, random())
....:     lp = g.longest_path()
....:     if (not lp.is_forest() or
....:         not max(lp.degree()) <= 2 or
....:         not lp.is_connected()):
....:         print("Error!")
....:         break

loop_edges(labels=True)

Return a list of all loops in the (di)graph

INPUT:

• labels – whether returned edges have labels ((u,v,l)) or not ((u,v))

EXAMPLES:

sage: G = Graph(loops=True); G
Looped graph on 0 vertices
sage: G.has_loops()
False
sage: G.allows_loops()
True
sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: G.loop_edges()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]
sage: G.loop_edges(labels=False)
[(0, 0), (1, 1), (2, 2), (3, 3)]
sage: G.allows_loops()
True
sage: G.has_loops()
True
sage: G.allow_loops(False)
sage: G.has_loops()
False
sage: G.loop_edges()
[]
sage: G.edges()
[(2, 3, None)]

sage: D = DiGraph(loops=True); D
Looped digraph on 0 vertices
sage: D.has_loops()
False
sage: D.allows_loops()
True
sage: D.add_edge((0,0))
sage: D.has_loops()
True
sage: D.loops()
[(0, 0, None)]
sage: D.allow_loops(False); D
Digraph on 1 vertex
sage: D.has_loops()
False
sage: D.edges()
[]

sage: G = graphs.PetersenGraph()
sage: G.loops()
[]

sage: D = DiGraph(4, loops=True)
sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: D.loop_edges()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]

sage: G = Graph(4, loops=True, multiedges=True, sparse=True)
sage: G.add_edges([(i,i) for i in range(4)])
sage: G.loop_edges()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]
sage: G.add_edges([(0, 0), (1, 1)])
sage: G.loop_edges(labels=False)
[(0, 0), (0, 0), (1, 1), (1, 1), (2, 2), (3, 3)]

loop_vertices()

Return a list of vertices with loops

EXAMPLES:

sage: G = Graph({0 : [0], 1: [1,2,3], 2: [3]}, loops=True)
sage: G.loop_vertices()
[0, 1]

loops(labels=True)

Return a list of all loops in the (di)graph

INPUT:

• labels – whether returned edges have labels ((u,v,l)) or not ((u,v))

EXAMPLES:

sage: G = Graph(loops=True); G
Looped graph on 0 vertices
sage: G.has_loops()
False
sage: G.allows_loops()
True
sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: G.loop_edges()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]
sage: G.loop_edges(labels=False)
[(0, 0), (1, 1), (2, 2), (3, 3)]
sage: G.allows_loops()
True
sage: G.has_loops()
True
sage: G.allow_loops(False)
sage: G.has_loops()
False
sage: G.loop_edges()
[]
sage: G.edges()
[(2, 3, None)]

sage: D = DiGraph(loops=True); D
Looped digraph on 0 vertices
sage: D.has_loops()
False
sage: D.allows_loops()
True
sage: D.add_edge((0,0))
sage: D.has_loops()
True
sage: D.loops()
[(0, 0, None)]
sage: D.allow_loops(False); D
Digraph on 1 vertex
sage: D.has_loops()
False
sage: D.edges()
[]

sage: G = graphs.PetersenGraph()
sage: G.loops()
[]

sage: D = DiGraph(4, loops=True)
sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: D.loop_edges()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]

sage: G = Graph(4, loops=True, multiedges=True, sparse=True)
sage: G.add_edges([(i,i) for i in range(4)])
sage: G.loop_edges()
[(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]
sage: G.add_edges([(0, 0), (1, 1)])
sage: G.loop_edges(labels=False)
[(0, 0), (0, 0), (1, 1), (1, 1), (2, 2), (3, 3)]

max_cut(value_only=True, use_edge_labels=False, vertices=False, solver=None, verbose=0)

Returns a maximum edge cut of the graph. For more information, see the Wikipedia article on cuts.

INPUT:

• value_only – boolean (default: True)
  • When set to True (default), only the value is returned.
  • When set to False, both the value and a maximum edge cut are returned.
• use_edge_labels – boolean (default: False)
  • When set to True, computes a maximum weighted cut where each edge has a weight defined by its label. (If an edge has no label, \(1\) is assumed.)
  • When set to False, each edge has weight \(1\).
• vertices – boolean (default: False)
  • When set to True, also returns the two sets of vertices that are disconnected by the cut. This implies value_only=False.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

EXAMPLES:

Quite obviously, the max cut of a bipartite graph is the number of edges, and the two sets of vertices are the two sides

sage: g = graphs.CompleteBipartiteGraph(5,6)
sage: [ value, edges, [ setA, setB ]] = g.max_cut(vertices=True)
sage: value == 5*6
True
sage: bsetA, bsetB  = map(list,g.bipartite_sets())
sage: (bsetA == setA and bsetB == setB ) or ((bsetA == setB and bsetB == setA ))
True

The max cut of a Petersen graph:

sage: g=graphs.PetersenGraph()
sage: g.max_cut()
12

merge_vertices(vertices)

Merge vertices.

This function replaces a set \(S\) of vertices by a single vertex \(v_{new}\), such that the edge \(uv_{new}\) exists if and only if \(\exists v'\in S: (u,v')\in G\).

The new vertex is named after the first vertex in the list given in argument. If this first name is \(None\), a new vertex is created.

In the case of multigraphs, the multiplicity is preserved.

INPUT:

• vertices – the list of vertices to be merged.

Note

If u and v are distinct vertices in vertices, any edges between u and v will be lost.

EXAMPLES:

sage: g=graphs.CycleGraph(3)
sage: g.merge_vertices([0,1])
sage: g.edges()
[(0, 2, None)]

sage: # With a Multigraph :
sage: g=graphs.CycleGraph(3)
sage: g.allow_multiple_edges(True)
sage: g.merge_vertices([0,1])
sage: g.edges(labels=False)
[(0, 2), (0, 2)]

sage: P=graphs.PetersenGraph()
sage: P.merge_vertices([5,7])
sage: P.vertices()
[0, 1, 2, 3, 4, 5, 6, 8, 9]

sage: g=graphs.CycleGraph(5)
sage: g.vertices()
[0, 1, 2, 3, 4]
sage: g.merge_vertices([None, 1, 3])
sage: g.edges(labels=False)
[(0, 4), (0, 5), (2, 5), (4, 5)]

min_spanning_tree(weight_function=None, algorithm='Prim_Boost', starting_vertex=None, check=False)

Returns the edges of a minimum spanning tree.

At the moment, no algorithm for directed graph is implemented: if the graph is directed, a minimum spanning tree of the corresponding undirected graph is returned.

We expect all weights of the graph to be convertible to float. Otherwise, an exception is raised.

INPUT:

• weight_function (function) - a function that inputs an edge e and outputs its weight. An edge has the form (u,v,l), where u and v are vertices, l is a label (that can be of any kind). The weight_function can be used to transform the label into a weight (note that, if the weight returned is not convertible to a float, an error is raised). In particular:
  • if weight_function is not None, the weight of an edge e is weight_function(e);
  • if weight_function is None (default) and g is weighted (that is, g.weighted()==True), for each edge e=(u,v,l), we set weight l;
  • if weight_function is None and g is not weighted, we set all weights to 1 (hence, the output can be any spanning tree).
• algorithm – The algorithm to use in computing a minimum spanning tree of G. The following algorithms are supported:
  • "Prim_Boost" (default) – Prim’s algorithm (Boost implementation).
  • "Prim_fringe" – a variant of Prim’s algorithm. "Prim_fringe" ignores the labels on the edges.
  • "Prim_edge" – a variant of Prim’s algorithm.
  • "Kruskal" – Kruskal’s algorithm.
  • "Kruskal_Boost" – Kruskal’s algorithm (Boost implementation).
  • NetworkX – Uses NetworkX’s minimum spanning tree implementation.
• starting_vertex – The vertex from which to begin the search for a minimum spanning tree (available only for Prim_fringe and Prim_edge).
• check – Boolean; default: False. Whether to first perform sanity checks on the input graph G. If appropriate, check is passed on to any minimum spanning tree functions that are invoked from the current method. See the documentation of the corresponding functions for details on what sort of sanity checks will be performed.

OUTPUT:

The edges of a minimum spanning tree of G, if one exists, otherwise returns the empty list.

See also

• sage.graphs.spanning_tree.kruskal()
• sage.graphs.base.boost_graph.min_spanning_tree()

EXAMPLES:

Kruskal’s algorithm:

sage: g = graphs.CompleteGraph(5)
sage: len(g.min_spanning_tree())
4
sage: weight = lambda e: 1 / ((e[0] + 1) * (e[1] + 1))
sage: g.min_spanning_tree(weight_function=weight)
[(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
sage: g.min_spanning_tree(weight_function=weight, algorithm='Kruskal_Boost')
[(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
sage: g = graphs.PetersenGraph()
sage: g.allow_multiple_edges(True)
sage: g.add_edges(g.edges())
sage: g.min_spanning_tree()
[(0, 1, None), (0, 4, None), (0, 5, None), (1, 2, None), (1, 6, None), (3, 8, None), (5, 7, None), (5, 8, None), (6, 9, None)]

Prim’s algorithm:

sage: g = graphs.CompleteGraph(5)
sage: g.min_spanning_tree(algorithm='Prim_edge', starting_vertex=2, weight_function=weight)
[(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
sage: g.min_spanning_tree(algorithm='Prim_fringe', starting_vertex=2, weight_function=weight)
[(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
sage: g.min_spanning_tree(weight_function=weight, algorithm='Prim_Boost')
[(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]

NetworkX algorithm:

sage: g.min_spanning_tree(algorithm='NetworkX')
[(0, 1, None), (0, 2, None), (0, 3, None), (0, 4, None)]

More complicated weights:

sage: G = Graph([(0,1,{'name':'a','weight':1}), (0,2,{'name':'b','weight':3}), (1,2,{'name':'b','weight':1})])
sage: G.min_spanning_tree(weight_function=lambda e: e[2]['weight'])
[(0, 1, {'name': 'a', 'weight': 1}), (1, 2, {'name': 'b', 'weight': 1})]

If the graph is not weighted, edge labels are not considered, even if they are numbers:

sage: g = Graph([[1,2,1], [1,3,2], [2,3,1]])
sage: g.min_spanning_tree()
[(1, 2, 1), (1, 3, 2)]

In order to use weights, we need to set variable weighted to True:

sage: g.weighted(True)
sage: g.min_spanning_tree()
[(1, 2, 1), (2, 3, 1)]

multicommodity_flow(terminals, integer=True, use_edge_labels=False, vertex_bound=False, solver=None, verbose=0)

Solves a multicommodity flow problem.

In the multicommodity flow problem, we are given a set of pairs \((s_i, t_i)\), called terminals meaning that \(s_i\) is willing some flow to \(t_i\).

Even though it is a natural generalisation of the flow problem this version of it is NP-Complete to solve when the flows are required to be integer.

For more information, see the Wikipedia page on multicommodity flows.

INPUT:

• terminals – a list of pairs \((s_i, t_i)\) or triples \((s_i, t_i, w_i)\) representing a flow from \(s_i\) to \(t_i\) of intensity \(w_i\). When the pairs are of size \(2\), a intensity of \(1\) is assumed.
• integer (boolean) – whether to require an integer multicommodity flow
• use_edge_labels (boolean) – whether to consider the label of edges as numerical values representing a capacity. If set to False, a capacity of \(1\) is assumed
• vertex_bound (boolean) – whether to require that a vertex can stand at most \(1\) commodity of flow through it of intensity \(1\). Terminals can obviously still send or receive several units of flow even though vertex_bound is set to True, as this parameter is meant to represent topological properties.
• solver – Specify a Linear Program solver to be used. If set to None, the default one is used. function of MixedIntegerLinearProgram. See the documentation of MixedIntegerLinearProgram.solve for more informations.
• verbose (integer) – sets the level of verbosity. Set to 0 by default (quiet).

ALGORITHM:

(Mixed Integer) Linear Program, depending on the value of integer.

EXAMPLES:

An easy way to obtain a satisfiable multiflow is to compute a matching in a graph, and to consider the paired vertices as terminals

sage: g = graphs.PetersenGraph()
sage: matching = [(u,v) for u,v,_ in g.matching()]
sage: h = g.multicommodity_flow(matching)
sage: len(h)
5

We could also have considered g as symmetric and computed the multiflow in this version instead. In this case, however edges can be used in both directions at the same time:

sage: h = DiGraph(g).multicommodity_flow(matching)
sage: len(h)
5

An exception is raised when the problem has no solution

sage: h = g.multicommodity_flow([(u,v,3) for u,v in matching])
Traceback (most recent call last):
...
EmptySetError: The multiflow problem has no solution

multiple_edges(to_undirected=False, labels=True)

Returns any multiple edges in the (di)graph.

EXAMPLES:

sage: G = Graph(multiedges=True,sparse=True); G
Multi-graph on 0 vertices
sage: G.has_multiple_edges()
False
sage: G.allows_multiple_edges()
True
sage: G.add_edges([(0,1)]*3)
sage: G.has_multiple_edges()
True
sage: G.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: G.allow_multiple_edges(False); G
Graph on 2 vertices
sage: G.has_multiple_edges()
False
sage: G.edges()
[(0, 1, None)]

sage: D = DiGraph(multiedges=True,sparse=True); D
Multi-digraph on 0 vertices
sage: D.has_multiple_edges()
False
sage: D.allows_multiple_edges()
True
sage: D.add_edges([(0,1)]*3)
sage: D.has_multiple_edges()
True
sage: D.multiple_edges()
[(0, 1, None), (0, 1, None), (0, 1, None)]
sage: D.allow_multiple_edges(False); D
Digraph on 2 vertices
sage: D.has_multiple_edges()
False
sage: D.edges()
[(0, 1, None)]

sage: G = DiGraph({1:{2: 'h'}, 2:{1:'g'}},sparse=True)
sage: G.has_multiple_edges()
False
sage: G.has_multiple_edges(to_undirected=True)
True
sage: G.multiple_edges()
[]
sage: G.multiple_edges(to_undirected=True)
[(1, 2, 'h'), (2, 1, 'g')]

multiway_cut(vertices, value_only=False, use_edge_labels=False, solver=None, verbose=0)

Returns a minimum edge multiway cut corresponding to the given set of vertices ( cf. http://www.d.kth.se/~viggo/wwwcompendium/node92.html ) represented by a list of edges.

A multiway cut for a vertex set \(S\) in a graph or a digraph \(G\) is a set \(C\) of edges such that any two vertices \(u,v\) in \(S\) are disconnected when removing the edges from \(C\) from \(G\).

Such a cut is said to be minimum when its cardinality (or weight) is minimum.

INPUT:

• vertices (iterable)– the set of vertices

• value_only (boolean)

  • When set to True, only the value of a minimum multiway cut is returned.
  • When set to False (default), the list of edges is returned

• use_edge_labels (boolean)

  • When set to True, computes a weighted minimum cut where each edge has a weight defined by its label. ( if an edge has no label, \(1\) is assumed )
  • when set to False (default), each edge has weight \(1\).

• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

EXAMPLES:

Of course, a multiway cut between two vertices correspond to a minimum edge cut

sage: g = graphs.PetersenGraph()
sage: g.edge_cut(0,3) == g.multiway_cut([0,3], value_only = True)
True

As Petersen’s graph is \(3\)-regular, a minimum multiway cut between three vertices contains at most \(2\times 3\) edges (which could correspond to the neighborhood of 2 vertices):

sage: g.multiway_cut([0,3,9], value_only = True) == 2*3
True

In this case, though, the vertices are an independent set. If we pick instead vertices \(0,9,\) and \(7\), we can save \(4\) edges in the multiway cut

sage: g.multiway_cut([0,7,9], value_only = True) == 2*3 - 1
True

This example, though, does not work in the directed case anymore, as it is not possible in Petersen’s graph to mutualise edges

sage: g = DiGraph(g)
sage: g.multiway_cut([0,7,9], value_only = True) == 3*3
True

Of course, a multiway cut between the whole vertex set contains all the edges of the graph:

sage: C = g.multiway_cut(g.vertices())
sage: set(C) == set(g.edges())
True

name(new=None)

Returns or sets the graph’s name.

INPUT:

• new - if not None, then this becomes the new name of the (di)graph. (if new == ‘’, removes any name)

EXAMPLES:

sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7, 8], 6: [8,9], 7: [9]}
sage: G = Graph(d); G
Graph on 10 vertices
sage: G.name("Petersen Graph"); G
Petersen Graph: Graph on 10 vertices
sage: G.name(new=""); G
Graph on 10 vertices
sage: G.name()
''

Name of an immutable graph trac ticket #15681

sage: g = graphs.PetersenGraph()
sage: gi = g.copy(immutable=True)
sage: gi.name()
'Petersen graph'
sage: gi.name("Hey")
Traceback (most recent call last):
...
NotImplementedError: An immutable graph does not change name

neighbor_iterator(vertex)

Return an iterator over neighbors of vertex.

EXAMPLES:

sage: G = graphs.CubeGraph(3)
sage: for i in G.neighbor_iterator('010'):
....:  print(i)
011
000
110
sage: D = G.to_directed()
sage: for i in D.neighbor_iterator('010'):
....:  print(i)
011
000
110

sage: D = DiGraph({0:[1,2], 3:[0]})
sage: list(D.neighbor_iterator(0))
[1, 2, 3]

neighbors(vertex)

Return a list of neighbors (in and out if directed) of vertex.

G[vertex] also works.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: sorted(P.neighbors(3))
[2, 4, 8]
sage: sorted(P[4])
[0, 3, 9]

networkx_graph(copy=True)

Creates a new NetworkX graph from the Sage graph.

INPUT:

• copy - if False, and the underlying implementation is a NetworkX graph, then the actual object itself is returned.

EXAMPLES:

sage: G = graphs.TetrahedralGraph()
sage: N = G.networkx_graph()
sage: type(N)
<class 'networkx.classes.graph.Graph'>

num_edges()

Returns the number of edges.

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.size()
15

num_faces(embedding=None)

Returns the number of faces of an embedded graph.

EXAMPLES:

sage: T = graphs.TetrahedralGraph()
sage: T.num_faces()
4

num_verts()

Returns the number of vertices. Note that len(G) returns the number of vertices in G also.

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.order()
10

sage: G = graphs.TetrahedralGraph()
sage: len(G)
4

number_of_loops()

Return the number of edges that are loops

EXAMPLES:

sage: G = Graph(4, loops=True)
sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: G.edges(labels=False)
[(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)]
sage: G.number_of_loops()
4

sage: D = DiGraph(4, loops=True)
sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: D.edges(labels=False)
[(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)]
sage: D.number_of_loops()
4

order()

Returns the number of vertices. Note that len(G) returns the number of vertices in G also.

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.order()
10

sage: G = graphs.TetrahedralGraph()
sage: len(G)
4

periphery(by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Returns the set of vertices in the periphery, i.e. whose eccentricity is equal to the diameter of the (di)graph.

In other words, the periphery is the set of vertices achieving the maximum eccentricity.

For more information and examples on how to use input variables, see shortest_paths() and eccentricity()

INPUT:

• by_weight - if True, edge weights are taken into account; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - a Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' for unweighted graphs, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost', otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

EXAMPLES:

sage: G = graphs.DiamondGraph()
sage: G.periphery()
[0, 3]
sage: P = graphs.PetersenGraph()
sage: P.subgraph(P.periphery()) == P
True
sage: S = graphs.StarGraph(19)
sage: S.periphery()
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: G = Graph()
sage: G.periphery()
[]
sage: G.add_vertex()
0
sage: G.periphery()
[0]

plot(**options)

Returns a graphics object representing the (di)graph.

INPUT:

• pos - an optional positioning dictionary

• layout - what kind of layout to use, takes precedence over pos

  • ‘circular’ – plots the graph with vertices evenly distributed on a circle
  • ‘spring’ - uses the traditional spring layout, using the graph’s current positions as initial positions
  • ‘tree’ - the (di)graph must be a tree. One can specify the root of the tree using the keyword tree_root, otherwise a root will be selected at random. Then the tree will be plotted in levels, depending on minimum distance for the root.

• vertex_labels - whether to print vertex labels

• edge_labels - whether to print edge labels. By default, False, but if True, the result of str(l) is printed on the edge for each label l. Labels equal to None are not printed (to set edge labels, see set_edge_label).

• edge_labels_background - The color of the edge labels background. The default is “white”. To achieve a transparent background use “transparent”.

• vertex_size - size of vertices displayed

• vertex_shape - the shape to draw the vertices, for example "o" for circle or "s" for square. Whole list is available at http://matplotlib.org/api/markers_api.html. (Not available for multiedge digraphs.)

• graph_border - whether to include a box around the graph

• vertex_colors - optional dictionary to specify vertex colors: each key is a color recognizable by matplotlib, and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it doesn’t get drawn).

• edge_colors - a dictionary specifying edge colors: each key is a color recognized by matplotlib, and each entry is a list of edges.

• partition - a partition of the vertex set. if specified, plot will show each cell in a different color. vertex_colors takes precedence.

• talk - if true, prints large vertices with white backgrounds so that labels are legible on slides

• iterations - how many iterations of the spring layout algorithm to go through, if applicable

• color_by_label - a boolean or dictionary or function (default: False) whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible with edge_color and edge_colors.

• heights - if specified, this is a dictionary from a set of floating point heights to a set of vertices

• edge_style - keyword arguments passed into the edge-drawing routine. This currently only works for directed graphs, since we pass off the undirected graph to networkx

• tree_root - a vertex of the tree to be used as the root for the layout=”tree” option. If no root is specified, then one is chosen at random. Ignored unless layout=’tree’.

• tree_orientation - “up” or “down” (default is “down”). If “up” (resp., “down”), then the root of the tree will appear on the bottom (resp., top) and the tree will grow upwards (resp. downwards). Ignored unless layout=’tree’.

• save_pos - save position computed during plotting

Note

• This method supports any parameter accepted by sage.plot.graphics.Graphics.show().
• See the documentation of the sage.graphs.graph_plot module for information and examples of how to define parameters that will be applied to all graph plots.
• Default parameters for this method and a specific graph can also be set through the options mechanism. For more information on this different way to set default parameters, see the help of the options decorator.
• See also the sage.graphs.graph_latex module for ways to use LaTeX to produce an image of a graph.

EXAMPLES:

sage: from sage.graphs.graph_plot import graphplot_options
sage: list(sorted(graphplot_options.items()))
[...]

sage: from math import sin, cos, pi
sage: P = graphs.PetersenGraph()
sage: d = {'#FF0000':[0,5], '#FF9900':[1,6], '#FFFF00':[2,7], '#00FF00':[3,8], '#0000FF':[4,9]}
sage: pos_dict = {}
sage: for i in range(5):
....:  x = float(cos(pi/2 + ((2*pi)/5)*i))
....:  y = float(sin(pi/2 + ((2*pi)/5)*i))
....:  pos_dict[i] = [x,y]
...
sage: for i in range(5, 10):
....:  x = float(0.5*cos(pi/2 + ((2*pi)/5)*i))
....:  y = float(0.5*sin(pi/2 + ((2*pi)/5)*i))
....:  pos_dict[i] = [x,y]
...
sage: pl = P.plot(pos=pos_dict, vertex_colors=d)
sage: pl.show()

sage: C = graphs.CubeGraph(8)
sage: P = C.plot(vertex_labels=False, vertex_size=0, graph_border=True)
sage: P.show()

sage: G = graphs.HeawoodGraph()
sage: for u,v,l in G.edges():
....:     G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
sage: G.plot(edge_labels=True).show()

sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []} , sparse=True)
sage: for u,v,l in D.edges():
....:     D.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
sage: D.plot(edge_labels=True, layout='circular').show()

sage: from sage.plot.colors import rainbow
sage: C = graphs.CubeGraph(5)
sage: R = rainbow(5)
sage: edge_colors = {}
sage: for i in range(5):
....:  edge_colors[R[i]] = []
sage: for u,v,l in C.edges():
....:  for i in range(5):
....:      if u[i] != v[i]:
....:          edge_colors[R[i]].append((u,v,l))
sage: C.plot(vertex_labels=False, vertex_size=0, edge_colors=edge_colors).show()

sage: D = graphs.DodecahedralGraph()
sage: Pi = [[6,5,15,14,7],[16,13,8,2,4],[12,17,9,3,1],[0,19,18,10,11]]
sage: D.show(partition=Pi)

sage: G = graphs.PetersenGraph()
sage: G.allow_loops(True)
sage: G.add_edge(0,0)
sage: G.show()

sage: D = DiGraph({0:[0,1], 1:[2], 2:[3]}, loops=True)
sage: D.show()
sage: D.show(edge_colors={(0,1,0):[(0,1,None),(1,2,None)],(0,0,0):[(2,3,None)]})

sage: pos = {0:[0.0, 1.5], 1:[-0.8, 0.3], 2:[-0.6, -0.8], 3:[0.6, -0.8], 4:[0.8, 0.3]}
sage: g = Graph({0:[1], 1:[2], 2:[3], 3:[4], 4:[0]})
sage: g.plot(pos=pos, layout='spring', iterations=0)
Graphics object consisting of 11 graphics primitives

sage: G = Graph()
sage: P = G.plot()
sage: P.axes()
False
sage: G = DiGraph()
sage: P = G.plot()
sage: P.axes()
False

sage: G = graphs.PetersenGraph()
sage: G.get_pos()
{0: (6.12..., 1.0...),
 1: (-0.95..., 0.30...),
 2: (-0.58..., -0.80...),
 3: (0.58..., -0.80...),
 4: (0.95..., 0.30...),
 5: (1.53..., 0.5...),
 6: (-0.47..., 0.15...),
 7: (-0.29..., -0.40...),
 8: (0.29..., -0.40...),
 9: (0.47..., 0.15...)}
sage: P = G.plot(save_pos=True, layout='spring')

The following illustrates the format of a position dictionary.

sage: G.get_pos() # currently random across platforms, see #9593
{0: [1.17..., -0.855...],
 1: [1.81..., -0.0990...],
 2: [1.35..., 0.184...],
 3: [1.51..., 0.644...],
 4: [2.00..., -0.507...],
 5: [0.597..., -0.236...],
 6: [2.04..., 0.687...],
 7: [1.46..., -0.473...],
 8: [0.902..., 0.773...],
 9: [2.48..., -0.119...]}

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]})
Graphics object consisting of 14 graphics primitives

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]})
Graphics object consisting of 14 graphics primitives
sage: t.set_edge_label(0,1,-7)
sage: t.set_edge_label(0,5,3)
sage: t.set_edge_label(0,5,99)
sage: t.set_edge_label(1,2,1000)
sage: t.set_edge_label(3,2,'spam')
sage: t.set_edge_label(2,6,3/2)
sage: t.set_edge_label(0,4,66)
sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}, edge_labels=True)
Graphics object consisting of 20 graphics primitives

sage: T = list(graphs.trees(7))
sage: t = T[3]
sage: t.plot(layout='tree')
Graphics object consisting of 14 graphics primitives

sage: t = DiGraph('[email protected]??GO??CO??GO??')
sage: t.plot(layout='tree', tree_root=0, tree_orientation="up")
Graphics object consisting of 22 graphics primitives
sage: D = DiGraph({0:[1,2,3], 2:[1,4], 3:[0]})
sage: D.plot()
Graphics object consisting of 16 graphics primitives

sage: D = DiGraph(multiedges=True,sparse=True)
sage: for i in range(5):
....:   D.add_edge((i,i+1,'a'))
....:   D.add_edge((i,i-1,'b'))
sage: D.plot(edge_labels=True,edge_colors=D._color_by_label())
Graphics object consisting of 34 graphics primitives
sage: D.plot(edge_labels=True, color_by_label={'a':'blue', 'b':'red'}, edge_style='dashed')
Graphics object consisting of 34 graphics primitives

sage: g = Graph({}, loops=True, multiedges=True,sparse=True)
sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'),
....:   (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')])
sage: g.plot(edge_labels=True, color_by_label=True, edge_style='dashed')
Graphics object consisting of 26 graphics primitives

sage: S = SupersingularModule(389)
sage: H = S.hecke_matrix(2)
sage: D = DiGraph(H,sparse=True)
sage: P = D.plot()

sage: G=Graph({'a':['a','b','b','b','e'],'b':['c','d','e'],'c':['c','d','d','d'],'d':['e']},sparse=True)
sage: G.show(pos={'a':[0,1],'b':[1,1],'c':[2,0],'d':[1,0],'e':[0,0]})

plot3d(bgcolor=(1, 1, 1), vertex_colors=None, vertex_size=0.06, vertex_labels=False, edge_colors=None, edge_size=0.02, edge_size2=0.0325, pos3d=None, color_by_label=False, engine='jmol', **kwds)

Plot a graph in three dimensions.

See also the sage.graphs.graph_latex module for ways to use LaTeX to produce an image of a graph.

INPUT:

• bgcolor - rgb tuple (default: (1,1,1))
• vertex_size - float (default: 0.06)
• vertex_labels – a boolean (default: False) whether to display vertices using text labels instead of spheres
• vertex_colors - optional dictionary to specify vertex colors: each key is a color recognizable by tachyon (rgb tuple (default: (1,0,0))), and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it doesn’t get drawn).
• edge_colors - a dictionary specifying edge colors: each key is a color recognized by tachyon ( default: (0,0,0) ), and each entry is a list of edges.
• color_by_label - a boolean or dictionary or function (default: False) whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible with edge_color and edge_colors.
• edge_size - float (default: 0.02)
• edge_size2 - float (default: 0.0325), used for Tachyon sleeves
• pos3d - a position dictionary for the vertices
• layout, iterations, ... - layout options; see layout()
• engine - which renderer to use. Options:
  • 'jmol' - default
  • 'tachyon'
• xres - resolution
• yres - resolution
• **kwds - passed on to the rendering engine

EXAMPLES:

sage: G = graphs.CubeGraph(5)
sage: G.plot3d(iterations=500, edge_size=None, vertex_size=0.04) # long time
Graphics3d Object

We plot a fairly complicated Cayley graph:

sage: A5 = AlternatingGroup(5); A5
Alternating group of order 5!/2 as a permutation group
sage: G = A5.cayley_graph()
sage: G.plot3d(vertex_size=0.03, edge_size=0.01, vertex_colors={(1,1,1):G.vertices()}, bgcolor=(0,0,0), color_by_label=True, iterations=200) # long time
Graphics3d Object

Some Tachyon examples:

sage: D = graphs.DodecahedralGraph()
sage: P3D = D.plot3d(engine='tachyon')
sage: P3D.show() # long time

sage: G = graphs.PetersenGraph()
sage: G.plot3d(engine='tachyon', vertex_colors={(0,0,1):G.vertices()}).show() # long time

sage: C = graphs.CubeGraph(4)
sage: C.plot3d(engine='tachyon', edge_colors={(0,1,0):C.edges()}, vertex_colors={(1,1,1):C.vertices()}, bgcolor=(0,0,0)).show() # long time

sage: K = graphs.CompleteGraph(3)
sage: K.plot3d(engine='tachyon', edge_colors={(1,0,0):[(0,1,None)], (0,1,0):[(0,2,None)], (0,0,1):[(1,2,None)]}).show() # long time

A directed version of the dodecahedron

sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []} )
sage: D.plot3d().show() # long time

sage: P = graphs.PetersenGraph().to_directed()
sage: from sage.plot.colors import rainbow
sage: edges = P.edges()
sage: R = rainbow(len(edges), 'rgbtuple')
sage: edge_colors = {}
sage: for i in range(len(edges)):
....:     edge_colors[R[i]] = [edges[i]]
sage: P.plot3d(engine='tachyon', edge_colors=edge_colors).show() # long time

sage: G=Graph({'a':['a','b','b','b','e'],'b':['c','d','e'],'c':['c','d','d','d'],'d':['e']},sparse=True)
sage: G.show3d()
Traceback (most recent call last):
...
NotImplementedError: 3D plotting of multiple edges or loops not implemented.

See also

• plot()
• graphviz_string()

radius(by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Returns the radius of the (di)graph.

The radius is defined to be the minimum eccentricity of any vertex, where the eccentricity is the maximum distance to any other vertex. For more information and examples on how to use input variables, see shortest_paths() and eccentricity()

INPUT:

• by_weight - if True, edge weights are taken into account; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - a Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' for unweighted graphs, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost', otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

EXAMPLES: The more symmetric a graph is, the smaller (diameter - radius) is.

sage: G = graphs.BarbellGraph(9, 3)
sage: G.radius()
3
sage: G.diameter()
6

sage: G = graphs.OctahedralGraph()
sage: G.radius()
2
sage: G.diameter()
2

random_edge(**kwds)

Returns a random edge of self.

INPUT:

• **kwds - arguments to be passed down to the edge_iterator method.

EXAMPLES:

The returned value is an edge of self:

sage: g = graphs.PetersenGraph()
sage: u,v = g.random_edge(labels=False)
sage: g.has_edge(u,v)
True

As the edges() method would, this function returns by default a triple (u,v,l) of values, in which l is the label of edge \((u,v)\):

sage: g.random_edge()
(3, 4, None)

random_edge_iterator(*args, **kwds)

Return an iterator over random edges of self.

The returned iterator enables to amortize the cost of accessing random edges, as can be done with multiple calls to method random_edge().

INPUT:

• *args and **kwds - arguments to be passed down to the edge_iterator() method.

EXAMPLES:

The returned value is an iterator over the edges of self:

sage: g = graphs.PetersenGraph()
sage: it = g.random_edge_iterator()
sage: [g.has_edge(next(it)) for _ in range(5)]
[True, True, True, True, True]

As the edges() method would, this function returns by default a triple (u,v,l) of values, in which l is the label of edge (u,v):

sage: print(next(g.random_edge_iterator())) # random
(0, 5, None)
sage: print(next(g.random_edge_iterator(labels=False))) # random
(5, 7)

random_subgraph(p, inplace=False)

Return a random subgraph that contains each vertex with prob. p.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.random_subgraph(.25)
Subgraph of (Petersen graph): Graph on 4 vertices

random_vertex(**kwds)

Returns a random vertex of self.

INPUT:

• **kwds - arguments to be passed down to the vertex_iterator method.

EXAMPLES:

The returned value is a vertex of self:

sage: g = graphs.PetersenGraph()
sage: v = g.random_vertex()
sage: v in g
True

random_vertex_iterator(*args, **kwds)

Return an iterator over random vertices of self.

The returned iterator enables to amortize the cost of accessing random vertices, as can be done with multiple calls to method random_vertex().

INPUT:

• *args and **kwds - arguments to be passed down to the vertex_iterator() method.

EXAMPLES:

The returned value is an iterator over the vertices of self:

sage: g = graphs.PetersenGraph()
sage: it = g.random_vertex_iterator()
sage: [next(it) in g for _ in range(5)]
[True, True, True, True, True]

relabel(perm=None, inplace=True, return_map=False, check_input=True, complete_partial_function=True, immutable=None)

Relabels the vertices of self

INPUT:

• perm – a function, dictionary, list, permutation, or None (default: None)
• inplace – a boolean (default: True)
• return_map – a boolean (default: False)
• check_input (boolean) – whether to test input for correctness. This can potentially be very time-consuming !.
• complete_partial_function (boolean) – whether to automatically complete the permutation if some elements of the graph are not associated with any new name. In this case, those elements are not relabeled This can potentially be very time-consuming !.
• immutable (boolean) – with inplace=False, whether to create a mutable/immutable relabelled copy. immutable=None (default) means that the graph and its copy will behave the same way.

If perm is a function f, then each vertex v is relabeled to f(v).

If perm is a dictionary d, then each vertex v (which should be a key of d) is relabeled to d[v]. Similarly, if perm is a list or tuple l of length n, then the first vertex returned by G.vertices() is relabeled to l[0], the second to l[1], ...

If perm is a permutation, then each vertex v is relabeled to perm(v). Caveat: this assumes that the vertices are labelled \(\{0,1,...,n-1\}\); since permutations act by default on the set \(\{1,2,...,n\}\), this is achieved by identifying \(n\) and \(0\).

If perm is None, the graph is relabeled to be on the vertices \(\{0,1,...,n-1\}\). This is not any kind of canonical labeling, but neither a random relabeling.

If inplace is True, the graph is modified in place and None is returned. Otherwise a relabeled copy of the graph is returned.

If return_map is True a dictionary representing the relabelling map is returned (incompatible with inplace==False).

EXAMPLES:

sage: G = graphs.PathGraph(3)
sage: G.am()
[0 1 0]
[1 0 1]
[0 1 0]

Relabeling using a dictionary. Note that the dictionary does not define the new label of vertex \(0\):

sage: G.relabel({1:2,2:1}, inplace=False).am()
[0 0 1]
[0 0 1]
[1 1 0]

This is because the method automatically “extends” the relabeling to the missing vertices (whose label will not change). Checking that all vertices have an image can require some time, and this feature can be disabled (at your own risk):

sage: G.relabel({1:2,2:1}, inplace=False, complete_partial_function = False).am()
Traceback (most recent call last):
...
KeyError: 0

Relabeling using a list:

sage: G.relabel([0,2,1], inplace=False).am()
[0 0 1]
[0 0 1]
[1 1 0]

Relabeling using a tuple:

sage: G.relabel((0,2,1), inplace=False).am()
[0 0 1]
[0 0 1]
[1 1 0]

Relabeling using a Sage permutation:

sage: G = graphs.PathGraph(3)
sage: from sage.groups.perm_gps.permgroup_named import SymmetricGroup
sage: S = SymmetricGroup(3)
sage: gamma = S('(1,2)')
sage: G.relabel(gamma, inplace=False).am()
[0 0 1]
[0 0 1]
[1 1 0]

A way to get a random relabeling:

sage: set_random_seed(0)  # Results are reproducible
sage: D = DiGraph({1: [2], 3: [4]})
sage: D.relabel(Permutations(D.vertices()).random_element())
sage: D.sources()
[1, 4]

Relabeling using an injective function:

sage: G.edges()
[(0, 1, None), (1, 2, None)]
sage: H = G.relabel(lambda i: i+10, inplace=False)
sage: H.vertices()
[10, 11, 12]
sage: H.edges()
[(10, 11, None), (11, 12, None)]

Relabeling using a non injective function has no meaning:

sage: G.edges()
[(0, 1, None), (1, 2, None)]
sage: G.relabel(lambda i: 0, inplace=False)
Traceback (most recent call last):
...
NotImplementedError: Non injective relabeling

But this test can be disabled, which can lead to ... problems:

sage: G.edges()
[(0, 1, None), (1, 2, None)]
sage: G.relabel(lambda i: 0, check_input = False)
sage: G.edges()
[]

Relabeling to simpler labels:

sage: G = graphs.CubeGraph(3)
sage: G.vertices()
['000', '001', '010', '011', '100', '101', '110', '111']
sage: G.relabel()
sage: G.vertices()
[0, 1, 2, 3, 4, 5, 6, 7]

Recovering the relabeling with return_map:

sage: G = graphs.CubeGraph(3)
sage: expecting = {'000': 0, '001': 1, '010': 2, '011': 3, '100': 4, '101': 5, '110': 6, '111': 7}
sage: G.relabel(return_map=True) == expecting
True

sage: G = graphs.PathGraph(3)
sage: G.relabel(lambda i: i+10, return_map=True)
{0: 10, 1: 11, 2: 12}

sage: G = graphs.PathGraph(5)
sage: G.set_vertices({0: 'before', 1: 'delete', 2: 'after'})
sage: G.delete_vertex(1)
sage: G.relabel()
sage: G.get_vertices()
{0: 'before', 1: 'after', 2: None, 3: None}
sage: G.get_pos()
{0: (0, 0), 1: (2, 0), 2: (3, 0), 3: (4, 0)}

Check that trac ticket #12477 is fixed:

sage: g = Graph({1:[2,3]})
sage: rel = {1:'a', 2:'b'}
sage: g.relabel(rel)
sage: g.vertices()
[3, 'a', 'b']
sage: rel
{1: 'a', 2: 'b'}

Immutable graphs cannot be relabeled:

sage: Graph(graphs.PetersenGraph(), immutable=True).relabel({})
Traceback (most recent call last):
...
ValueError: To relabel an immutable graph use inplace=False

trac ticket #16257:

sage: G = graphs.PetersenGraph()
sage: G.relabel( [ i+1 for i in range(G.order()) ], inplace=True )
sage: G.relabel( [ i+1 for i in range(G.order()) ], inplace=True )

remove_loops(vertices=None)

Removes loops on vertices in vertices. If vertices is None, removes all loops.

EXAMPLE

sage: G = Graph(4, loops=True)
sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: G.edges(labels=False)
[(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)]
sage: G.remove_loops()
sage: G.edges(labels=False)
[(2, 3)]
sage: G.allows_loops()
True
sage: G.has_loops()
False

sage: D = DiGraph(4, loops=True)
sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] )
sage: D.edges(labels=False)
[(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)]
sage: D.remove_loops()
sage: D.edges(labels=False)
[(2, 3)]
sage: D.allows_loops()
True
sage: D.has_loops()
False

remove_multiple_edges()

Removes all multiple edges, retaining one edge for each.

EXAMPLES:

sage: G = Graph(multiedges=True, sparse=True)
sage: G.add_edges( [ (0,1), (0,1), (0,1), (0,1), (1,2) ] )
sage: G.edges(labels=False)
[(0, 1), (0, 1), (0, 1), (0, 1), (1, 2)]

sage: G.remove_multiple_edges()
sage: G.edges(labels=False)
[(0, 1), (1, 2)]

sage: D = DiGraph(multiedges=True, sparse=True)
sage: D.add_edges( [ (0,1,1), (0,1,2), (0,1,3), (0,1,4), (1,2,None) ] )
sage: D.edges(labels=False)
[(0, 1), (0, 1), (0, 1), (0, 1), (1, 2)]
sage: D.remove_multiple_edges()
sage: D.edges(labels=False)
[(0, 1), (1, 2)]

set_edge_label(u, v, l)

Set the edge label of a given edge.

Note

There can be only one edge from u to v for this to make sense. Otherwise, an error is raised.

INPUT:

• u, v - the vertices (and direction if digraph) of the edge
• l - the new label

EXAMPLES:

sage: SD = DiGraph( { 1:[18,2], 2:[5,3], 3:[4,6], 4:[7,2], 5:[4], 6:[13,12], 7:[18,8,10], 8:[6,9,10], 9:[6], 10:[11,13], 11:[12], 12:[13], 13:[17,14], 14:[16,15], 15:[2], 16:[13], 17:[15,13], 18:[13] }, sparse=True)
sage: SD.set_edge_label(1, 18, 'discrete')
sage: SD.set_edge_label(4, 7, 'discrete')
sage: SD.set_edge_label(2, 5, 'h = 0')
sage: SD.set_edge_label(7, 18, 'h = 0')
sage: SD.set_edge_label(7, 10, 'aut')
sage: SD.set_edge_label(8, 10, 'aut')
sage: SD.set_edge_label(8, 9, 'label')
sage: SD.set_edge_label(8, 6, 'no label')
sage: SD.set_edge_label(13, 17, 'k > h')
sage: SD.set_edge_label(13, 14, 'k = h')
sage: SD.set_edge_label(17, 15, 'v_k finite')
sage: SD.set_edge_label(14, 15, 'v_k m.c.r.')
sage: posn = {1:[ 3,-3],  2:[0,2],  3:[0, 13],  4:[3,9],  5:[3,3],  6:[16, 13], 7:[6,1],  8:[6,6],  9:[6,11], 10:[9,1], 11:[10,6], 12:[13,6], 13:[16,2], 14:[10,-6], 15:[0,-10], 16:[14,-6], 17:[16,-10], 18:[6,-4]}
sage: SD.plot(pos=posn, vertex_size=400, vertex_colors={'#FFFFFF':list(range(1,19))}, edge_labels=True).show() # long time

sage: G = graphs.HeawoodGraph()
sage: for u,v,l in G.edges():
....:  G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')')
sage: G.edges()
    [(0, 1, '(0,1)'),
     (0, 5, '(0,5)'),
     (0, 13, '(0,13)'),
     ...
     (11, 12, '(11,12)'),
     (12, 13, '(12,13)')]

sage: g = Graph({0: [0,1,1,2]}, loops=True, multiedges=True, sparse=True)
sage: g.set_edge_label(0,0,'test')
sage: g.edges()
[(0, 0, 'test'), (0, 1, None), (0, 1, None), (0, 2, None)]
sage: g.add_edge(0,0,'test2')
sage: g.set_edge_label(0,0,'test3')
Traceback (most recent call last):
...
RuntimeError: Cannot set edge label, since there are multiple edges from 0 to 0.

sage: dg = DiGraph({0 : [1], 1 : [0]}, sparse=True)
sage: dg.set_edge_label(0,1,5)
sage: dg.set_edge_label(1,0,9)
sage: dg.outgoing_edges(1)
[(1, 0, 9)]
sage: dg.incoming_edges(1)
[(0, 1, 5)]
sage: dg.outgoing_edges(0)
[(0, 1, 5)]
sage: dg.incoming_edges(0)
[(1, 0, 9)]

sage: G = Graph({0:{1:1}}, sparse=True)
sage: G.num_edges()
1
sage: G.set_edge_label(0,1,1)
sage: G.num_edges()
1

set_embedding(embedding)

Sets a combinatorial embedding dictionary to _embedding attribute.

Dictionary is organized with vertex labels as keys and a list of each vertex’s neighbors in clockwise order.

Dictionary is error-checked for validity.

INPUT:

• embedding - a dictionary

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.set_embedding({0: [1, 5, 4], 1: [0, 2, 6], 2: [1, 3, 7], 3: [8, 2, 4], 4: [0, 9, 3], 5: [0, 8, 7], 6: [8, 1, 9], 7: [9, 2, 5], 8: [3, 5, 6], 9: [4, 6, 7]})
sage: G.set_embedding({'s': [1, 5, 4], 1: [0, 2, 6], 2: [1, 3, 7], 3: [8, 2, 4], 4: [0, 9, 3], 5: [0, 8, 7], 6: [8, 1, 9], 7: [9, 2, 5], 8: [3, 5, 6], 9: [4, 6, 7]})
Traceback (most recent call last):
...
ValueError: The following vertices from the embedding do not belong to the graph: ['s']

set_latex_options(**kwds)

Sets multiple options for rendering a graph with LaTeX.

INPUT:

• kwds - any number of option/value pairs to set many graph latex options at once (a variable number, in any order). Existing values are overwritten, new values are added. Existing values can be cleared by setting the value to None. Possible options are documented at sage.graphs.graph_latex.GraphLatex.set_option().

This method is a convenience for setting the options of a graph directly on an instance of the graph. For a full explanation of how to use LaTeX to render graphs, see the introduction to the graph_latex module.

EXAMPLES:

sage: g = graphs.PetersenGraph()
sage: g.set_latex_options(tkz_style = 'Welsh')
sage: opts = g.latex_options()
sage: opts.get_option('tkz_style')
'Welsh'

set_planar_positions(test=False, **layout_options)

Compute a planar layout for self using Schnyder’s algorithm, and save it as default layout.

EXAMPLES:

sage: g = graphs.CycleGraph(7)
sage: g.set_planar_positions(test=True)
True

This method is deprecated since Sage-4.4.1.alpha2. Please use instead:

sage: g.layout(layout = “planar”, save_pos = True) {0: [1, 1], 1: [2, 2], 2: [3, 2], 3: [1, 4], 4: [5, 1], 5: [0, 5], 6: [1, 0]}

set_pos(pos, dim=2)

Sets the position dictionary, a dictionary specifying the coordinates of each vertex.

EXAMPLES: Note that set_pos will allow you to do ridiculous things, which will not blow up until plotting:

sage: G = graphs.PetersenGraph()
sage: G.get_pos()
{0: (..., ...),
 ...
 9: (..., ...)}

sage: G.set_pos('spam')
sage: P = G.plot()
Traceback (most recent call last):
...
TypeError: string indices must be integers, not str

set_vertex(vertex, object)

Associate an arbitrary object with a vertex.

INPUT:

• vertex - which vertex
• object - object to associate to vertex

EXAMPLES:

sage: T = graphs.TetrahedralGraph()
sage: T.vertices()
[0, 1, 2, 3]
sage: T.set_vertex(1, graphs.FlowerSnark())
sage: T.get_vertex(1)
Flower Snark: Graph on 20 vertices

set_vertices(vertex_dict)

Associate arbitrary objects with each vertex, via an association dictionary.

INPUT:

• vertex_dict - the association dictionary

EXAMPLES:

sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() }
sage: d[2]
Moebius-Kantor Graph: Graph on 16 vertices
sage: T = graphs.TetrahedralGraph()
sage: T.vertices()
[0, 1, 2, 3]
sage: T.set_vertices(d)
sage: T.get_vertex(1)
Flower Snark: Graph on 20 vertices

shortest_path(u, v, by_weight=False, algorithm=None, weight_function=None, check_weight=True, bidirectional=None)

Returns a list of vertices representing some shortest path from u to v: if there is no path from u to v, the list is empty.

For more information and more examples, see shortest_paths() (the inputs are very similar).

INPUT:

• u, v (vertices) - the start and the end vertices of the paths.

• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.

• algorithm (string) - one of the following algorithms:

  • 'BFS': performs a BFS from u. Does not work with edge weights.
  • 'BFS_Bid: performs a BFS from u and from v. Does not work with edge weights.
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. Works only with positive weights.
  • 'Dijkstra_Bid_NetworkX': performs a Dijkstra visit from u and from v (NetworkX implementation). Works only with positive weights.
  • 'Dijkstra_Bid': a Cython implementation that performs a Dijkstra visit from u and from v. Works only with positive weights.
  • 'Bellman-Ford_Boost': the Bellman-Ford algorithm, implemented in Boost. Works also with negative weights, if there is no negative cycle.
  • None (default): Sage chooses the best algorithm: 'BFS_Bid' if by_weight is False, 'Dijkstra_Bid' otherwise.

  Note

  If there are negative weights and algorithm is None, the result is not reliable. This occurs because, for performance reasons, we cannot check whether there are edges with negative weights before running the algorithm. If there are, the user should explicitly input algorithm='Bellman-Ford_Boost'.

• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.

• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

• bidirectional - Deprecated and replaced by Algorithm: now it has no effect. Before, if it was True, the algorithm would expand vertices from u and v at the same time, making two spheres of half the usual radius.

EXAMPLES:

sage: D = graphs.DodecahedralGraph()
sage: D.shortest_path(4, 9)
[4, 17, 16, 12, 13, 9]
sage: D.shortest_path(4, 9, algorithm='BFS')
[4, 3, 2, 1, 8, 9]
sage: D.shortest_path(4, 8, algorithm='Dijkstra_NetworkX')
[4, 3, 2, 1, 8]
sage: D.shortest_path(4, 8, algorithm='Dijkstra_Bid_NetworkX')
[4, 3, 2, 1, 8]
sage: D.shortest_path(4, 9, algorithm='Dijkstra_Bid')
[4, 3, 19, 0, 10, 9]
sage: D.shortest_path(5, 5)
[5]
sage: D.delete_edges(D.edges_incident(13))
sage: D.shortest_path(13, 4)
[]
sage: G = Graph({0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True)
sage: G.plot(edge_labels=True).show() # long time
sage: G.shortest_path(0, 3)
[0, 4, 3]
sage: G.shortest_path(0, 3, by_weight=True)
[0, 1, 2, 3]
sage: G.shortest_path(0, 3, by_weight=True, algorithm='Dijkstra_NetworkX')
[0, 1, 2, 3]
sage: G.shortest_path(0, 3, by_weight=True, algorithm='Dijkstra_Bid_NetworkX')
[0, 1, 2, 3]

shortest_path_all_pairs(by_weight=False, algorithm=None, weight_function=None, check_weight=True, default_weight=None)

Computes a shortest path between each pair of vertices.

INPUT:

• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS' - the computation is done through a BFS centered on each vertex successively. Works only if by_weight==False.
  • 'Floyd-Warshall-Cython' - the Cython implementation of the Floyd-Warshall algorithm. Works only if by_weight==False.
  • 'Floyd-Warshall-Python' - the Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Johnson_Boost': the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' if by_weight is False, 'Dijkstra_Boost' if all weights are positive, 'Floyd-Warshall-Cython' otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.
• default_weight - Deprecated: now it has no effect. Before, it was used to assign a weight to edges with no label. Now it has been replaced by weight_function.

OUTPUT:

A tuple (dist, pred). They are both dicts of dicts. The first indicates the length dist[u][v] of the shortest weighted path from \(u\) to \(v\). The second is a compact representation of all the paths - it indicates the predecessor pred[u][v] of \(v\) in the shortest path from \(u\) to \(v\). If the algorithm used is Johnson_Boost, predecessors are not computed.

Note

Only reachable vertices are present in the dictionaries.

Note

There is a Cython version of this method that is usually much faster for large graphs, as most of the time is actually spent building the final double dictionary. Everything on the subject is to be found in the distances_all_pairs module.

EXAMPLES:

Some standard examples (see shortest_paths() for more examples on how to use the input variables):

sage: G = Graph( { 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True)
sage: G.plot(edge_labels=True).show() # long time
sage: dist, pred = G.shortest_path_all_pairs(by_weight = True)
sage: dist
{0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 3}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 3}, 3: {0: 3, 1: 2, 2: 1, 3: 0, 4: 2}, 4: {0: 2, 1: 3, 2: 3, 3: 2, 4: 0}}
sage: pred
{0: {0: None, 1: 0, 2: 1, 3: 2, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 1, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}}
sage: pred[0]
{0: None, 1: 0, 2: 1, 3: 2, 4: 0}
sage: G = Graph( { 0: {1: {'weight':1}}, 1: {2: {'weight':1}}, 2: {3: {'weight':1}}, 3: {4: {'weight':2}}, 4: {0: {'weight':2}} }, sparse=True)
sage: dist, pred = G.shortest_path_all_pairs(weight_function = lambda e:e[2]['weight'])
sage: dist
{0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 3}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 3}, 3: {0: 3, 1: 2, 2: 1, 3: 0, 4: 2}, 4: {0: 2, 1: 3, 2: 3, 3: 2, 4: 0}}
sage: pred
{0: {0: None, 1: 0, 2: 1, 3: 2, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 1, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}}

So for example the shortest weighted path from \(0\) to \(3\) is obtained as follows. The predecessor of \(3\) is pred[0][3] == 2, the predecessor of \(2\) is pred[0][2] == 1, and the predecessor of \(1\) is pred[0][1] == 0.

sage: G = Graph( { 0: {1:None}, 1: {2:None}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True )
sage: G.shortest_path_all_pairs()
({0: {0: 0, 1: 1, 2: 2, 3: 2, 4: 1},
1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 2},
2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 2},
3: {0: 2, 1: 2, 2: 1, 3: 0, 4: 1},
4: {0: 1, 1: 2, 2: 2, 3: 1, 4: 0}},
{0: {0: None, 1: 0, 2: 1, 3: 4, 4: 0},
1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0},
2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3},
3: {0: 4, 1: 2, 2: 3, 3: None, 4: 3},
4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}})
sage: G.shortest_path_all_pairs(weight_function=lambda e:(e[2] if e[2] is not None else 1))
({0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 2},
1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 3},
2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 3},
3: {0: 3, 1: 2, 2: 1, 3: 0, 4: 2},
4: {0: 2, 1: 3, 2: 3, 3: 2, 4: 0}},
{0: {0: None, 1: 0, 2: 1, 3: 2, 4: 0},
1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0},
2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3},
3: {0: 1, 1: 2, 2: 3, 3: None, 4: 3},
4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}})

Now, default_weight does not work anymore:

sage: G.shortest_path_all_pairs(by_weight = True, default_weight=200)
Traceback (most recent call last):
...
ValueError: The weight function cannot find the weight of (0, 1, None).

It can be replaced by choosing an appropriate weight_function:

sage: G.shortest_path_all_pairs(weight_function=lambda e:(e[2] if e[2] is not None else 200))
({0: {0: 0, 1: 200, 2: 5, 3: 4, 4: 2},
1: {0: 200, 1: 0, 2: 200, 3: 201, 4: 202},
2: {0: 5, 1: 200, 2: 0, 3: 1, 4: 3},
3: {0: 4, 1: 201, 2: 1, 3: 0, 4: 2},
4: {0: 2, 1: 202, 2: 3, 3: 2, 4: 0}},
{0: {0: None, 1: 0, 2: 3, 3: 4, 4: 0},
1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0},
2: {0: 4, 1: 2, 2: None, 3: 2, 4: 3},
3: {0: 4, 1: 2, 2: 3, 3: None, 4: 3},
4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}})

Checking that distances are equal regardless of the algorithm used:

sage: g = graphs.Grid2dGraph(5,5)
sage: d1, _ = g.shortest_path_all_pairs(algorithm="BFS")
sage: d2, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Cython")
sage: d3, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Python")
sage: d4, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_NetworkX")
sage: d5, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_Boost")
sage: d6, _ = g.shortest_path_all_pairs(algorithm="Johnson_Boost")
sage: d1 == d2 == d3 == d4 == d5 == d6
True

Checking that distances are equal regardless of the algorithm used:

sage: g = digraphs.RandomDirectedGNM(6,12)
sage: d1, _ = g.shortest_path_all_pairs(algorithm="BFS")
sage: d2, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Cython")
sage: d3, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Python")
sage: d4, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_NetworkX")
sage: d5, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_Boost")
sage: d6, _ = g.shortest_path_all_pairs(algorithm="Johnson_Boost")
sage: d1 == d2 == d3 == d4 == d5 == d6
True

Checking that weighted distances are equal regardless of the algorithm used:

sage: g = Graph()
sage: import random
sage: for v in range(5):
....:     for w in range(5):
....:         if v != w:
....:             g.add_edge(v, w, random.uniform(1,10))
sage: d1, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Python")
sage: d2, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_NetworkX")
sage: d3, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_Boost")
sage: d4, _ = g.shortest_path_all_pairs(algorithm="Johnson_Boost")
sage: d1 == d2 == d3 == d4
True

Checking a random path is valid:

sage: dist, path = g.shortest_path_all_pairs(algorithm="BFS")
sage: u,v = g.random_vertex(), g.random_vertex()
sage: p = [v]
sage: while p[0] is not None:
....:   p.insert(0,path[u][p[0]])
sage: len(p) == dist[u][v] + 2
True

Negative weights:

sage: g = DiGraph([(0,1,-2),(1,0,1)], weighted=True)
sage: g.shortest_path_all_pairs(by_weight=True)
Traceback (most recent call last):
...
ValueError: The graph contains a negative cycle.

Unreachable vertices are not present in the dictionaries:

sage: g = DiGraph([(0,1,1),(1,2,2)])
sage: g.shortest_path_all_pairs(algorithm='BFS')
({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}},
 {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}})
sage: g.shortest_path_all_pairs(algorithm='Dijkstra_NetworkX')
({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}},
 {0: {0: None, 1: 1, 2: 1}, 1: {1: None, 2: 2}, 2: {2: None}})
sage: g.shortest_path_all_pairs(algorithm='Dijkstra_Boost')
({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}},
 {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}})
sage: g.shortest_path_all_pairs(algorithm='Floyd-Warshall-Python')
({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}},
 {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}})
sage: g.shortest_path_all_pairs(algorithm='Floyd-Warshall-Cython')
({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}},
 {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}})

In order to change the default behavior if the graph is disconnected, we can use default values with dictionaries:

sage: G = 2*graphs.PathGraph(2)
sage: d,_ = G.shortest_path_all_pairs()
sage: import itertools
sage: from sage.rings.infinity import Infinity
sage: for u,v in itertools.combinations(G.vertices(),2):
....:     print("dist({}, {}) = {}".format(u,v, d[u].get(v,+Infinity)))
dist(0, 1) = 1
dist(0, 2) = +Infinity
dist(0, 3) = +Infinity
dist(1, 2) = +Infinity
dist(1, 3) = +Infinity
dist(2, 3) = 1

shortest_path_length(u, v, by_weight=False, algorithm=None, weight_function=None, check_weight=True, bidirectional=None, weight_sum=None)

Returns the minimal length of a path from u to v.

If there is no path from u to v, returns Infinity.

For more information and more examples, we refer to shortest_path() and shortest_paths(), which have very similar inputs.

INPUT:

• u, v (vertices) - the start and the end vertices of the paths.

• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.

• algorithm (string) - one of the following algorithms:

  • 'BFS': performs a BFS from u. Does not work with edge weights.
  • 'BFS_Bid: performs a BFS from u and from v. Does not work with edge weights.
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX.
  • 'Dijkstra_Bid_NetworkX': performs a Dijkstra visit from u and from v (NetworkX implementation).
  • 'Dijkstra_Bid': a Cython implementation that performs a Dijkstra visit from u and from v.
  • 'Bellman-Ford_Boost': the Bellman-Ford algorithm, implemented in Boost. Works also with negative weights, if there is no negative cycle.
  • None (default): Sage chooses the best algorithm: 'BFS_Bid' if by_weight is False, 'Dijkstra_Bid' otherwise.

  Note

  If there are negative weights and algorithm is None, the result is not reliable. This occurs because, for performance reasons, we cannot check whether there are edges with negative weights before running the algorithm. If there are, the user should explicitly input algorithm='Bellman-Ford_Boost'.

• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.

• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

• bidirectional - Deprecated and replaced by Algorithm: now it has no effect. Before, if it was True, the algorithm would expand vertices from u and v at the same time, making two spheres of half the usual radius.

• weight_sum - Deprecated: now it has no effect. Before, it was used to decide if the algorithm should return the number of edges in the shortest path or the length of the (weighted) path. Now it has the same value as by_weight.

EXAMPLES:

Standard examples:

sage: D = graphs.DodecahedralGraph()
sage: D.shortest_path_length(4, 9)
5
sage: D.shortest_path_length(4, 9, algorithm='BFS')
5
sage: D.shortest_path_length(4, 9, algorithm='Dijkstra_NetworkX')
5
sage: D.shortest_path_length(4, 9, algorithm='Dijkstra_Bid_NetworkX')
5
sage: D.shortest_path_length(4, 9, algorithm='Dijkstra_Bid')
5
sage: D.shortest_path_length(4, 9, algorithm='Bellman-Ford_Boost')
5
sage: D.shortest_path_length(5, 5)
0
sage: D.delete_edges(D.edges_incident(13))
sage: D.shortest_path_length(13, 4)
+Infinity
sage: G = Graph({0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True)
sage: G.plot(edge_labels=True).show() # long time
sage: G.shortest_path_length(0, 3)
2
sage: G.shortest_path_length(0, 3, by_weight=True)
3
sage: G.shortest_path_length(0, 3, by_weight=True, algorithm='Dijkstra_NetworkX')
3
sage: G.shortest_path_length(0, 3, by_weight=True, algorithm='Dijkstra_Bid_NetworkX')
3

If Dijkstra is used with negative weights, usually it raises an error:

sage: G = DiGraph({0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: -2}}, sparse = True)
sage: G.shortest_path_length(4, 1, by_weight=True, algorithm=None)
Traceback (most recent call last):
...
ValueError: The graph contains an edge with negative weight!
sage: G.shortest_path_length(4, 1, by_weight=True, algorithm='Bellman-Ford_Boost')
-1

However, sometimes the result may be wrong, and no error is raised:

sage: G = DiGraph([(0,1,1),(1,2,1),(0,3,1000),(3,4,-3000), (4,2,1000)])
sage: G.shortest_path_length(0, 2, by_weight=True, algorithm='Bellman-Ford_Boost')
-1000
sage: G.shortest_path_length(0, 2, by_weight=True)
2

shortest_path_lengths(u, by_weight=False, algorithm=None, weight_function=None, check_weight=True, weight_sums=None)

Computes the length of a shortest path from u to any other vertex.

Returns a dictionary of shortest path lengths keyed by targets, excluding all vertices that are not reachable from u.

For more information on the input variables and more examples, we refer to shortest_paths() which has the same input variables.

INPUT:

• u (vertex) - the starting vertex.
• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS': performs a BFS from u. Does not work with edge weights.
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX (works only with positive weights).
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Bellman-Ford_Boost': the Bellman-Ford algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' if by_weight is False, 'Dijkstra_Boost' if all weights are positive, 'Bellman-Ford_Boost' otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.
• weight_sums - Deprecated: now this variable has no effect. Before, it was used to decide whether the number of edges or the sum of their lengths was outputted. Now we use variable by_weight to decide.

EXAMPLES:

Unweighted case:

sage: D = graphs.DodecahedralGraph()
sage: D.shortest_path_lengths(0)
{0: 0, 1: 1, 2: 2, 3: 2, 4: 3, 5: 4, 6: 3, 7: 3, 8: 2, 9: 2, 10: 1, 11: 2, 12: 3, 13: 3, 14: 4, 15: 5, 16: 4, 17: 3, 18: 2, 19: 1}

Weighted case:

sage: G = Graph( { 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True)
sage: G.plot(edge_labels=True).show() # long time
sage: G.shortest_path_lengths(0, by_weight=True)
{0: 0, 1: 1, 2: 2, 3: 3, 4: 2}

Using a weight function:

sage: D = DiGraph([(0,1,{'weight':1}),(1,2,{'weight':3}),(0,2,{'weight':5})])
sage: weight_function = lambda e:e[2]['weight']
sage: D.shortest_path_lengths(1, algorithm='Dijkstra_NetworkX', by_weight=False)
{1: 0, 2: 1}
sage: D.shortest_path_lengths(0, weight_function=weight_function)
{0: 0, 1: 1, 2: 4}
sage: D.shortest_path_lengths(1, weight_function=weight_function)
{1: 0, 2: 3}

Negative weights:

sage: D = DiGraph([(0,1,{'weight':-1}),(1,2,{'weight':3}),(0,2,{'weight':5})])
sage: D.shortest_path_lengths(0, weight_function=weight_function)
{0: 0, 1: -1, 2: 2}

Negative cycles:

sage: D = DiGraph([(0,1,{'weight':-5}),(1,2,{'weight':3}),(2,0,{'weight':1})])
sage: D.shortest_path_lengths(0, weight_function=weight_function)
Traceback (most recent call last):
...
ValueError: the graph contains a negative cycle

Checking that distances are equal regardless of the algorithm used:

sage: g = graphs.Grid2dGraph(5,5)
sage: d1 = g.shortest_path_lengths((0,0), algorithm="BFS")
sage: d2 = g.shortest_path_lengths((0,0), algorithm="Dijkstra_NetworkX")
sage: d3 = g.shortest_path_lengths((0,0), algorithm="Dijkstra_Boost")
sage: d4 = g.shortest_path_lengths((0,0), algorithm="Bellman-Ford_Boost")
sage: d1 == d2 == d3 == d4
True

shortest_paths(u, by_weight=False, algorithm=None, weight_function=None, check_weight=True, cutoff=None)

Returns a dictionary associating to each vertex v a shortest path from u to v, if it exists.

If u and v are not connected, vertex v is not present in the dictionary.

INPUT:

• u (vertex) - the starting vertex.
• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - one of the following algorithms:
  • 'BFS': performs a BFS from u. Does not work with edge weights.
  • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX (works only with positive weights).
  • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost (works only with positive weights).
  • 'Bellman-Ford_Boost': the Bellman-Ford algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
  • None (default): Sage chooses the best algorithm: 'BFS' if by_weight is False, 'Dijkstra_Boost' if all weights are positive, 'Bellman-Ford_Boost' otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.
• cutoff (integer) - integer depth to stop search (used only if algorithm=='BFS').

EXAMPLES:

Standard example:

sage: D = graphs.DodecahedralGraph()
sage: D.shortest_paths(0)
{0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 19, 3], 4: [0, 19, 3, 4], 5: [0, 1, 2, 6, 5], 6: [0, 1, 2, 6], 7: [0, 1, 8, 7], 8: [0, 1, 8], 9: [0, 10, 9], 10: [0, 10], 11: [0, 10, 11], 12: [0, 10, 11, 12], 13: [0, 10, 9, 13], 14: [0, 1, 8, 7, 14], 15: [0, 19, 18, 17, 16, 15], 16: [0, 19, 18, 17, 16], 17: [0, 19, 18, 17], 18: [0, 19, 18], 19: [0, 19]}

All these paths are obviously induced graphs:

sage: all([D.subgraph(p).is_isomorphic(graphs.PathGraph(len(p)) )for p in D.shortest_paths(0).values()])
True

sage: D.shortest_paths(0, cutoff=2)
{0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 19, 3], 8: [0, 1, 8], 9: [0, 10, 9], 10: [0, 10], 11: [0, 10, 11], 18: [0, 19, 18], 19: [0, 19]}
sage: G = Graph( { 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True)
sage: G.plot(edge_labels=True).show() # long time
sage: G.shortest_paths(0, by_weight=True)
{0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 4]}

Weighted shortest paths:

sage: D = DiGraph([(0,1,1),(1,2,3),(0,2,5)])
sage: D.shortest_paths(0)
{0: [0], 1: [0, 1], 2: [0, 2]}
sage: D.shortest_paths(0, by_weight=True)
{0: [0], 1: [0, 1], 2: [0, 1, 2]}

Using a weight function (this way, by_weight is set to True):

sage: D = DiGraph([(0,1,{'weight':1}),(1,2,{'weight':3}),(0,2,{'weight':5})])
sage: weight_function = lambda e:e[2]['weight']
sage: D.shortest_paths(0, weight_function=weight_function)
{0: [0], 1: [0, 1], 2: [0, 1, 2]}

If the weight function does not match the label:

sage: D.shortest_paths(0, weight_function=lambda e:e[2])
Traceback (most recent call last):
...
ValueError: The weight function cannot find the weight of (0, 1, {'weight': 1}).

However, if check_weight is set to False, unexpected behavior may occur:

sage: D.shortest_paths(0, algorithm='Dijkstra_NetworkX', weight_function=lambda e:e[2], check_weight=False)
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'int' and 'dict'

Negative weights:

sage: D = DiGraph([(0,1,1),(1,2,-2),(0,2,4)])
sage: D.shortest_paths(0, by_weight=True)
{0: [0], 1: [0, 1], 2: [0, 1, 2]}

Negative cycles:

sage: D.add_edge(2,0,0)
sage: D.shortest_paths(0, by_weight=True)
Traceback (most recent call last):
...
ValueError: the graph contains a negative cycle

show(method='matplotlib', **kwds)

Shows the (di)graph.

INPUT:

• method – set to "matplotlib" (default) to display the graph using matplotlib, or to "js" to visualize it in a browser using d3.js.

• Any other argument supported by the drawing functions:

  • "matplotlib" – see GenericGraph.plot and sage.plot.graphics.Graphics.show().
  • "js" – see gen_html_code().

EXAMPLES:

sage: C = graphs.CubeGraph(8)
sage: P = C.plot(vertex_labels=False, vertex_size=0, graph_border=True)
sage: P.show()  # long time (3s on sage.math, 2011)

show3d(bgcolor=(1, 1, 1), vertex_colors=None, vertex_size=0.06, edge_colors=None, edge_size=0.02, edge_size2=0.0325, pos3d=None, color_by_label=False, engine='jmol', **kwds)

Plots the graph using Tachyon, and shows the resulting plot.

INPUT:

• bgcolor - rgb tuple (default: (1,1,1))
• vertex_size - float (default: 0.06)
• vertex_colors - optional dictionary to specify vertex colors: each key is a color recognizable by tachyon (rgb tuple (default: (1,0,0))), and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it doesn’t get drawn).
• edge_colors - a dictionary specifying edge colors: each key is a color recognized by tachyon ( default: (0,0,0) ), and each entry is a list of edges.
• edge_size - float (default: 0.02)
• edge_size2 - float (default: 0.0325), used for Tachyon sleeves
• pos3d - a position dictionary for the vertices
• iterations - how many iterations of the spring layout algorithm to go through, if applicable
• engine - which renderer to use. Options:
• 'jmol' - default ‘tachyon’
• xres - resolution
• yres - resolution
• **kwds - passed on to the Tachyon command

EXAMPLES:

sage: G = graphs.CubeGraph(5)
sage: G.show3d(iterations=500, edge_size=None, vertex_size=0.04) # long time

We plot a fairly complicated Cayley graph:

sage: A5 = AlternatingGroup(5); A5
Alternating group of order 5!/2 as a permutation group
sage: G = A5.cayley_graph()
sage: G.show3d(vertex_size=0.03, edge_size=0.01, edge_size2=0.02, vertex_colors={(1,1,1):G.vertices()}, bgcolor=(0,0,0), color_by_label=True, iterations=200) # long time

Some Tachyon examples:

sage: D = graphs.DodecahedralGraph()
sage: D.show3d(engine='tachyon') # long time

sage: G = graphs.PetersenGraph()
sage: G.show3d(engine='tachyon', vertex_colors={(0,0,1):G.vertices()}) # long time

sage: C = graphs.CubeGraph(4)
sage: C.show3d(engine='tachyon', edge_colors={(0,1,0):C.edges()}, vertex_colors={(1,1,1):C.vertices()}, bgcolor=(0,0,0)) # long time

sage: K = graphs.CompleteGraph(3)
sage: K.show3d(engine='tachyon', edge_colors={(1,0,0):[(0,1,None)], (0,1,0):[(0,2,None)], (0,0,1):[(1,2,None)]}) # long time

size()

Returns the number of edges.

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.size()
15

spanning_trees_count(root_vertex=None)

Returns the number of spanning trees in a graph.

In the case of a digraph, counts the number of spanning out-trees rooted in root_vertex. Default is to set first vertex as root.

This computation uses Kirchhoff’s Matrix Tree Theorem [1] to calculate the number of spanning trees. For complete graphs on \(n\) vertices the result can also be reached using Cayley’s formula: the number of spanning trees are \(n^(n-2)\).

For digraphs, the augmented Kirchhoff Matrix as defined in [2] is used for calculations. Here the result is the number of out-trees rooted at a specific vertex.

INPUT:

• root_vertex – integer (default: the first vertex) This is the vertex that will be used as root for all spanning out-trees if the graph is a directed graph. This argument is ignored if the graph is not a digraph.

See also

spanning_trees() – enumerates all spanning trees of a graph.

REFERENCES:

• [1] http://mathworld.wolfram.com/MatrixTreeTheorem.html
• [2] Lih-Hsing Hsu, Cheng-Kuan Lin, “Graph Theory and Interconnection Networks”

AUTHORS:

• Anders Jonsson (2009-10-10)

EXAMPLES:

sage: G = graphs.PetersenGraph()
sage: G.spanning_trees_count()
2000

sage: n = 11
sage: G = graphs.CompleteGraph(n)
sage: ST = G.spanning_trees_count()
sage: ST == n^(n-2)
True

sage: M=matrix(3,3,[0,1,0,0,0,1,1,1,0])
sage: D=DiGraph(M)
sage: D.spanning_trees_count()
1
sage: D.spanning_trees_count(0)
1
sage: D.spanning_trees_count(2)
2

spectral_radius(G, prec=1e-10)

Return an interval of floating point number that encloses the spectral radius of this graph

The input graph G must be strongly connected.

INPUT:

• prec – (default 1e-10) an upper bound for the relative precision of the interval

The algorithm is iterative and uses an inequality valid for non-negative matrices. Namely, if \(A\) is a non-negative square matrix with Perron-Frobenius eigenvalue \(\lambda\) then the following inequality is valid for any vector \(x\)

\[\min_i \frac{(Ax)_i}{x_i} \leq \lambda \leq \max_i \frac{(Ax)_i}{x_i}\]

Note

The speed of convergence of the algorithm is governed by the spectral gap (the distance to the second largest modulus of other eigenvalues). If this gap is small, then this function might not be appropriate.

The algorithm is not smart and not parallel! It uses basic interval arithmetic and native floating point arithmetic.

EXAMPLES:

sage: from sage.graphs.base.static_sparse_graph import spectral_radius

sage: G = DiGraph([(0,0),(0,1),(1,0)], loops=True)
sage: phi = (RR(1) + RR(5).sqrt() ) / 2
sage: phi  # abs tol 1e-14
1.618033988749895
sage: e_min, e_max = spectral_radius(G, 1e-14)
sage: e_min, e_max     # abs tol 1e-14
(1.618033988749894, 1.618033988749896)
sage: (e_max - e_min)  # abs tol 1e-14
1e-14
sage: e_min < phi < e_max
True

This function also works for graphs:

sage: G = Graph([(0,1),(0,2),(1,2),(1,3),(2,4),(3,4)])
sage: e_min, e_max = spectral_radius(G, 1e-14)
sage: e = max(G.adjacency_matrix().charpoly().roots(AA, multiplicities=False))
sage: e_min < e < e_max
True

sage: G.spectral_radius()  # abs tol 1e-9
(2.48119430408, 2.4811943041)

A larger example:

sage: G = DiGraph()
sage: G.add_edges((i,i+1) for i in range(200))
sage: G.add_edge(200,0)
sage: G.add_edge(1,0)
sage: e_min, e_max = spectral_radius(G, 0.00001)
sage: p = G.adjacency_matrix(sparse=True).charpoly()
sage: p
x^201 - x^199 - 1
sage: r = p.roots(AA, multiplicities=False)[0]
sage: e_min < r < e_max
True

A much larger example:

sage: G = DiGraph(100000)
sage: r = list(range(100000))
sage: while not G.is_strongly_connected():
....:     shuffle(r)
....:     G.add_edges(enumerate(r), loops=False)
sage: spectral_radius(G, 1e-10)  # random
(1.9997956006500042, 1.9998043797692782)

The algorithm takes care of multiple edges:

sage: G = DiGraph(2,loops=True,multiedges=True)
sage: G.add_edges([(0,0),(0,0),(0,1),(1,0)])
sage: spectral_radius(G, 1e-14)  # abs tol 1e-14
(2.414213562373094, 2.414213562373095)
sage: max(G.adjacency_matrix().eigenvalues(AA))
2.414213562373095?

spectrum(laplacian=False)

Returns a list of the eigenvalues of the adjacency matrix.

INPUT:

• laplacian - if True, use the Laplacian matrix (see kirchhoff_matrix())

OUTPUT:

A list of the eigenvalues, including multiplicities, sorted with the largest eigenvalue first.

See also

The method spectral_radius() returns floating point approximation of the maximum eigenvalue.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.spectrum()
[3, 1, 1, 1, 1, 1, -2, -2, -2, -2]
sage: P.spectrum(laplacian=True)
[5, 5, 5, 5, 2, 2, 2, 2, 2, 0]
sage: D = P.to_directed()
sage: D.delete_edge(7,9)
sage: D.spectrum()
[2.9032119259..., 1, 1, 1, 1, 0.8060634335..., -1.7092753594..., -2, -2, -2]

sage: C = graphs.CycleGraph(8)
sage: C.spectrum()
[2, 1.4142135623..., 1.4142135623..., 0, 0, -1.4142135623..., -1.4142135623..., -2]

A digraph may have complex eigenvalues. Previously, the complex parts of graph eigenvalues were being dropped. For a 3-cycle, we have:

sage: T = DiGraph({0:[1], 1:[2], 2:[0]})
sage: T.spectrum()
[1, -0.5000000000... + 0.8660254037...*I, -0.5000000000... - 0.8660254037...*I]

steiner_tree(vertices, weighted=False, solver=None, verbose=0)

Returns a tree of minimum weight connecting the given set of vertices.

Definition :

Computing a minimum spanning tree in a graph can be done in \(n \log(n)\) time (and in linear time if all weights are equal) where \(n = V + E\). On the other hand, if one is given a large (possibly weighted) graph and a subset of its vertices, it is NP-Hard to find a tree of minimum weight connecting the given set of vertices, which is then called a Steiner Tree.

Wikipedia article on Steiner Trees.

INPUT:

• vertices – the vertices to be connected by the Steiner Tree.
• weighted (boolean) – Whether to consider the graph as weighted, and use each edge’s label as a weight, considering None as a weight of \(1\). If weighted=False (default) all edges are considered to have a weight of \(1\).
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

Note

• This problem being defined on undirected graphs, the orientation is not considered if the current graph is actually a digraph.
• The graph is assumed not to have multiple edges.

ALGORITHM:

Solved through Linear Programming.

COMPLEXITY:

NP-Hard.

Note that this algorithm first checks whether the given set of vertices induces a connected graph, returning one of its spanning trees if weighted is set to False, and thus answering very quickly in some cases

EXAMPLES:

The Steiner Tree of the first 5 vertices in a random graph is, of course, always a tree

sage: g = graphs.RandomGNP(30,.5)
sage: st = g.steiner_tree(g.vertices()[:5])
sage: st.is_tree()
True

And all the 5 vertices are contained in this tree

sage: all([v in st for v in g.vertices()[:5] ])
True

An exception is raised when the problem is impossible, i.e. if the given vertices are not all included in the same connected component

sage: g = 2 * graphs.PetersenGraph()
sage: st = g.steiner_tree([5,15])
Traceback (most recent call last):
...
EmptySetError: The given vertices do not all belong to the same connected component. This problem has no solution !

strong_product(other)

Returns the strong product of self and other.

The strong product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)=V(G)\times V(H)\), and \(((u,v), (w,x))\) is an edge of \(L\) iff either :

• \((u, w)\) is an edge of \(G\) and \(v = x\), or
• \((v, x)\) is an edge of \(H\) and \(u = w\), or
• \((u, w)\) is an edge of \(G\) and \((v, x)\) is an edge of \(H\).

In other words, the edges of the strong product is the union of the edges of the tensor and Cartesian products.

EXAMPLES:

sage: Z = graphs.CompleteGraph(2)
sage: C = graphs.CycleGraph(5)
sage: S = C.strong_product(Z); S
Graph on 10 vertices
sage: S.plot() # long time
Graphics object consisting of 36 graphics primitives

sage: D = graphs.DodecahedralGraph()
sage: P = graphs.PetersenGraph()
sage: S = D.strong_product(P); S
Graph on 200 vertices
sage: S.plot() # long time
Graphics object consisting of 1701 graphics primitives

subdivide_edge(*args)

Subdivides an edge \(k\) times.

INPUT:

The following forms are all accepted to subdivide \(8\) times the edge between vertices \(1\) and \(2\) labeled with "my_label".

• G.subdivide_edge( 1, 2, 8 )
• G.subdivide_edge( (1, 2), 8 )
• G.subdivide_edge( (1, 2, "my_label"), 8 )

Note

• If the given edge is labelled with \(l\), all the edges created by the subdivision will have the same label.
• If no label is given, the label used will be the one returned by the method edge_label() on the pair u,v

EXAMPLES:

Subdividing \(5\) times an edge in a path of length \(3\) makes it a path of length \(8\):

sage: g = graphs.PathGraph(3)
sage: edge = g.edges()[0]
sage: g.subdivide_edge(edge, 5)
sage: g.is_isomorphic(graphs.PathGraph(8))
True

Subdividing a labelled edge in two ways

sage: g = Graph()
sage: g.add_edge(0,1,"label1")
sage: g.add_edge(1,2,"label2")
sage: print(sorted(g.edges()))
[(0, 1, 'label1'), (1, 2, 'label2')]

Specifying the label:

sage: g.subdivide_edge(0,1,"label1", 3)
sage: print(sorted(g.edges()))
[(0, 3, 'label1'), (1, 2, 'label2'), (1, 5, 'label1'), (3, 4, 'label1'), (4, 5, 'label1')]

The lazy way:

sage: g.subdivide_edge(1,2,"label2", 5)
sage: print(sorted(g.edges()))
[(0, 3, 'label1'), (1, 5, 'label1'), (1, 6, 'label2'), (2, 10, 'label2'), (3, 4, 'label1'), (4, 5, 'label1'), (6, 7, 'label2'), (7, 8, 'label2'), (8, 9, 'label2'), (9, 10, 'label2')]

If too many arguments are given, an exception is raised

sage: g.subdivide_edge(0,1,1,1,1,1,1,1,1,1,1)
Traceback (most recent call last):
...
ValueError: This method takes at most 4 arguments !

The same goes when the given edge does not exist:

sage: g.subdivide_edge(0,1,"fake_label",5)
Traceback (most recent call last):
...
ValueError: The given edge does not exist.

See also

• subdivide_edges() – subdivides multiples edges at a time

subdivide_edges(edges, k)

Subdivides k times edges from an iterable container.

For more information on the behaviour of this method, please refer to the documentation of subdivide_edge().

INPUT:

• edges – a list of edges
• k (integer) – common length of the subdivisions

Note

If a given edge is labelled with \(l\), all the edges created by its subdivision will have the same label.

EXAMPLES:

If we are given the disjoint union of several paths:

sage: paths = [2,5,9]
sage: paths = map(graphs.PathGraph, paths)
sage: g = Graph()
sage: for P in paths:
....:   g = g + P

... subdividing edges in each of them will only change their lengths:

sage: edges = [P.edges()[0] for P in g.connected_components_subgraphs()]
sage: k = 6
sage: g.subdivide_edges(edges, k)

Let us check this by creating the graph we expect to have built through subdivision:

sage: paths2 = [2+k, 5+k, 9+k]
sage: paths2 = map(graphs.PathGraph, paths2)
sage: g2 = Graph()
sage: for P in paths2:
....:   g2 = g2 + P
sage: g.is_isomorphic(g2)
True

See also

• subdivide_edge() – subdivides one edge

subgraph(vertices=None, edges=None, inplace=False, vertex_property=None, edge_property=None, algorithm=None, immutable=None)

Returns the subgraph containing the given vertices and edges.

If either vertices or edges are not specified, they are assumed to be all vertices or edges. If edges are not specified, returns the subgraph induced by the vertices.

INPUT:

• inplace - Using inplace is True will simply delete the extra vertices and edges from the current graph. This will modify the graph.
• vertices - Vertices can be a single vertex or an iterable container of vertices, e.g. a list, set, graph, file or numeric array. If not passed, defaults to the entire graph.
• edges - As with vertices, edges can be a single edge or an iterable container of edges (e.g., a list, set, file, numeric array, etc.). If not edges are not specified, then all edges are assumed and the returned graph is an induced subgraph. In the case of multiple edges, specifying an edge as (u,v) means to keep all edges (u,v), regardless of the label.
• vertex_property - If specified, this is expected to be a function on vertices, which is intersected with the vertices specified, if any are.
• edge_property - If specified, this is expected to be a function on edges, which is intersected with the edges specified, if any are.
• algorithm - If algorithm=delete or inplace=True, then the graph is constructed by deleting edges and vertices. If add, then the graph is constructed by building a new graph from the appropriate vertices and edges. If not specified, then the algorithm is chosen based on the number of vertices in the subgraph.
• immutable (boolean) – whether to create a mutable/immutable subgraph. immutable=None (default) means that the graph and its subgraph will behave the same way.

EXAMPLES:

sage: G = graphs.CompleteGraph(9)
sage: H = G.subgraph([0,1,2]); H
Subgraph of (Complete graph): Graph on 3 vertices
sage: G
Complete graph: Graph on 9 vertices
sage: J = G.subgraph(edges=[(0,1)])
sage: J.edges(labels=False)
[(0, 1)]
sage: J.vertices()==G.vertices()
True
sage: G.subgraph([0,1,2], inplace=True); G
Subgraph of (Complete graph): Graph on 3 vertices
sage: G.subgraph()==G
True

sage: D = graphs.CompleteGraph(9).to_directed()
sage: H = D.subgraph([0,1,2]); H
Subgraph of (Complete graph): Digraph on 3 vertices
sage: H = D.subgraph(edges=[(0,1), (0,2)])
sage: H.edges(labels=False)
[(0, 1), (0, 2)]
sage: H.vertices()==D.vertices()
True
sage: D
Complete graph: Digraph on 9 vertices
sage: D.subgraph([0,1,2], inplace=True); D
Subgraph of (Complete graph): Digraph on 3 vertices
sage: D.subgraph()==D
True

A more complicated example involving multiple edges and labels.

sage: G = Graph(multiedges=True, sparse=True)
sage: G.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')])
sage: G.subgraph(edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges()
[(0, 1, 'a'), (0, 1, 'b'), (0, 1, 'c'), (0, 2, 'd')]
sage: J = G.subgraph(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')])
sage: J.edges()
[(0, 1, 'a')]
sage: J.vertices()
[0, 1]
sage: G.subgraph(vertices=G.vertices())==G
True

sage: D = DiGraph(multiedges=True, sparse=True)
sage: D.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')])
sage: D.subgraph(edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges()
[(0, 1, 'a'), (0, 1, 'b'), (0, 2, 'd')]
sage: H = D.subgraph(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')])
sage: H.edges()
[(0, 1, 'a')]
sage: H.vertices()
[0, 1]

Using the property arguments:

sage: P = graphs.PetersenGraph()
sage: S = P.subgraph(vertex_property = lambda v : v%2 == 0)
sage: S.vertices()
[0, 2, 4, 6, 8]

sage: C = graphs.CubeGraph(2)
sage: S = C.subgraph(edge_property=(lambda e: e[0][0] == e[1][0]))
sage: C.edges()
[('00', '01', None), ('00', '10', None), ('01', '11', None), ('10', '11', None)]
sage: S.edges()
[('00', '01', None), ('10', '11', None)]

The algorithm is not specified, then a reasonable choice is made for speed.

sage: g=graphs.PathGraph(1000)
sage: g.subgraph(list(range(10))) # uses the 'add' algorithm
Subgraph of (Path graph): Graph on 10 vertices

sage: g = graphs.PathGraph(10)
sage: g.is_planar(set_embedding=True)
True
sage: g.set_vertices(dict((v, 'v%d'%v) for v in g.vertices()))
sage: h = g.subgraph([3..5])
sage: h.get_pos().keys()
[3, 4, 5]
sage: h.get_vertices()
{3: 'v3', 4: 'v4', 5: 'v5'}

subgraph_search(G, induced=False)

Returns a copy of G in self.

INPUT:

• G – the (di)graph whose copy we are looking for in self.
• induced – boolean (default: False). Whether or not to search for an induced copy of G in self.

OUTPUT:

• If induced=False, return a copy of G in this graph. Otherwise, return an induced copy of G in self. If G is the empty graph, return the empty graph since it is a subgraph of every graph. Now suppose G is not the empty graph. If there is no copy (induced or otherwise) of G in self, we return None.

Note

This method does not take vertex/edge labels into account.

See also

• subgraph_search_count() – Counts the number of copies of \(H\) inside of \(G\)
• subgraph_search_iterator() – Iterate on the copies of \(H\) inside of \(G\)

ALGORITHM:

See the documentation of SubgraphSearch.

EXAMPLES:

The Petersen graph contains the path graph \(P_5\):

sage: g = graphs.PetersenGraph()
sage: h1 = g.subgraph_search(graphs.PathGraph(5)); h1
Subgraph of (Petersen graph): Graph on 5 vertices
sage: h1.vertices(); h1.edges(labels=False)
[0, 1, 2, 3, 4]
[(0, 1), (1, 2), (2, 3), (3, 4)]
sage: I1 = g.subgraph_search(graphs.PathGraph(5), induced=True); I1
Subgraph of (Petersen graph): Graph on 5 vertices
sage: I1.vertices(); I1.edges(labels=False)
[0, 1, 2, 3, 8]
[(0, 1), (1, 2), (2, 3), (3, 8)]

It also contains the claw \(K_{1,3}\):

sage: h2 = g.subgraph_search(graphs.ClawGraph()); h2
Subgraph of (Petersen graph): Graph on 4 vertices
sage: h2.vertices(); h2.edges(labels=False)
[0, 1, 4, 5]
[(0, 1), (0, 4), (0, 5)]
sage: I2 = g.subgraph_search(graphs.ClawGraph(), induced=True); I2
Subgraph of (Petersen graph): Graph on 4 vertices
sage: I2.vertices(); I2.edges(labels=False)
[0, 1, 4, 5]
[(0, 1), (0, 4), (0, 5)]

Of course the induced copies are isomorphic to the graphs we were looking for:

sage: I1.is_isomorphic(graphs.PathGraph(5))
True
sage: I2.is_isomorphic(graphs.ClawGraph())
True

However, the Petersen graph does not contain a subgraph isomorphic to \(K_3\):

sage: g.subgraph_search(graphs.CompleteGraph(3)) is None
True

Nor does it contain a nonempty induced subgraph isomorphic to \(P_6\):

sage: g.subgraph_search(graphs.PathGraph(6), induced=True) is None
True

The empty graph is a subgraph of every graph:

sage: g.subgraph_search(graphs.EmptyGraph())
Graph on 0 vertices
sage: g.subgraph_search(graphs.EmptyGraph(), induced=True)
Graph on 0 vertices

The subgraph may just have edges missing:

sage: k3=graphs.CompleteGraph(3); p3=graphs.PathGraph(3)
sage: k3.relabel(list('abc'))
sage: s=k3.subgraph_search(p3)
sage: s.edges(labels=False)
[('a', 'b'), ('b', 'c')]

Of course, \(P_3\) is not an induced subgraph of \(K_3\), though:

sage: k3=graphs.CompleteGraph(3); p3=graphs.PathGraph(3)
sage: k3.relabel(list('abc'))
sage: k3.subgraph_search(p3, induced=True) is None
True

subgraph_search_count(G, induced=False)

Returns the number of labelled occurences of G in self.

INPUT:

• G – the (di)graph whose copies we are looking for in self.
• induced – boolean (default: False). Whether or not to count induced copies of G in self.

Note

This method does not take vertex/edge labels into account.

ALGORITHM:

See the documentation of SubgraphSearch.

See also

• subgraph_search() – finds an subgraph isomorphic to \(H\) inside of a graph \(G\)
• subgraph_search_iterator() – Iterate on the copies of a graph \(H\) inside of a graph \(G\)

EXAMPLES:

Counting the number of paths \(P_5\) in a PetersenGraph:

sage: g = graphs.PetersenGraph()
sage: g.subgraph_search_count(graphs.PathGraph(5))
240

Requiring these subgraphs be induced:

sage: g.subgraph_search_count(graphs.PathGraph(5), induced = True)
120

If we define the graph \(T_k\) (the transitive tournament on \(k\) vertices) as the graph on \(\{0, ..., k-1\}\) such that \(ij \in T_k\) iif \(i<j\), how many directed triangles can be found in \(T_5\) ? The answer is of course \(0\)

sage: T5 = DiGraph()
sage: T5.add_edges([(i,j) for i in range(5) for j in range(i+1, 5)])
sage: T5.subgraph_search_count(digraphs.Circuit(3))
0

If we count instead the number of \(T_3\) in \(T_5\), we expect the answer to be \(\binom{5}{3}\):

sage: T3 = T5.subgraph([0,1,2])
sage: T5.subgraph_search_count(T3)
10
sage: binomial(5,3)
10

The empty graph is a subgraph of every graph:

sage: g.subgraph_search_count(graphs.EmptyGraph())
1

subgraph_search_iterator(G, induced=False)

Returns an iterator over the labelled copies of G in self.

INPUT:

• G – the graph whose copies we are looking for in self.
• induced – boolean (default: False). Whether or not to iterate over the induced copies of G in self.

Note

This method does not take vertex/edge labels into account.

ALGORITHM:

See the documentation of SubgraphSearch.

OUTPUT:

Iterator over the labelled copies of G in self, as lists. For each value \((v_1, v_2, ..., v_k)\) returned, the first vertex of \(G\) is associated with \(v_1\), the second with \(v_2\), etc ...

Note

This method also works on digraphs.

See also

• subgraph_search() – finds an subgraph isomorphic to \(H\) inside of \(G\)
• subgraph_search_count() – Counts the number of copies of \(H\) inside of \(G\)

EXAMPLES:

Iterating through all the labelled \(P_3\) of \(P_5\):

sage: g = graphs.PathGraph(5)
sage: for p in g.subgraph_search_iterator(graphs.PathGraph(3)):
....:     print(p)
[0, 1, 2]
[1, 2, 3]
[2, 1, 0]
[2, 3, 4]
[3, 2, 1]
[4, 3, 2]

szeged_index()

Returns the Szeged index of the graph.

For any \(uv\in E(G)\), let \(N_u(uv) = \{w\in G:d(u,w)<d(v,w)\}, n_u(uv)=|N_u(uv)|\)

The Szeged index of a graph is then defined as [1]: \(\sum_{uv \in E(G)}n_u(uv)\times n_v(uv)\)

EXAMPLES:

True for any connected graph [1]:

sage: g=graphs.PetersenGraph()
sage: g.wiener_index()<= g.szeged_index()
True

True for all trees [1]:

sage: g=Graph()
sage: g.add_edges(graphs.CubeGraph(5).min_spanning_tree())
sage: g.wiener_index() == g.szeged_index()
True

REFERENCE:

[1] Klavzar S., Rajapakse A., Gutman I. (1996). The Szeged and the Wiener index of graphs. Applied Mathematics Letters, 9 (5), pp. 45-49.

tensor_product(other)

Returns the tensor product of self and other.

The tensor product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)\) equal to the Cartesian product of the vertices \(V(G)\) and \(V(H)\), and \(((u,v), (w,x))\) is an edge iff - \((u, w)\) is an edge of self, and - \((v, x)\) is an edge of other.

The tensor product is also known as the categorical product and the kronecker product (refering to the kronecker matrix product). See Wikipedia article on the Kronecker product.

EXAMPLES:

sage: Z = graphs.CompleteGraph(2)
sage: C = graphs.CycleGraph(5)
sage: T = C.tensor_product(Z); T
Graph on 10 vertices
sage: T.size()
10
sage: T.plot() # long time
Graphics object consisting of 21 graphics primitives

sage: D = graphs.DodecahedralGraph()
sage: P = graphs.PetersenGraph()
sage: T = D.tensor_product(P); T
Graph on 200 vertices
sage: T.size()
900
sage: T.plot() # long time
Graphics object consisting of 1101 graphics primitives

to_dictionary(edge_labels=False, multiple_edges=False)

Returns the graph as a dictionary.

INPUT:

• edge_labels (boolean) – whether to include edge labels in the output.
• multiple_edges (boolean) – whether to include multiple edges in the output.

OUTPUT:

The output depends on the input:

• If edge_labels == False and multiple_edges == False, the output is a dictionary associating to each vertex the list of its neighbors.
• If edge_labels == False and multiple_edges == True, the output is a dictionary the same as previously with one difference: the neighbors are listed with multiplicity.
• If edge_labels == True and multiple_edges == False, the output is a dictionary associating to each vertex \(u\) [a dictionary associating to each vertex \(v\) incident to \(u\) the label of edge \((u,v)\)].
• If edge_labels == True and multiple_edges == True, the output is a dictionary associating to each vertex \(u\) [a dictionary associating to each vertex \(v\) incident to \(u\) [the list of labels of all edges between \(u\) and \(v\)]].

Note

When used on directed graphs, the explanations above can be understood by replacing the word “neigbours” by “out-neighbors”

EXAMPLES:

sage: g = graphs.PetersenGraph().to_dictionary()
sage: [(key, sorted(g[key])) for key in g]
[(0, [1, 4, 5]),
 (1, [0, 2, 6]),
 (2, [1, 3, 7]),
 (3, [2, 4, 8]),
 (4, [0, 3, 9]),
 (5, [0, 7, 8]),
 (6, [1, 8, 9]),
 (7, [2, 5, 9]),
 (8, [3, 5, 6]),
 (9, [4, 6, 7])]
sage: graphs.PetersenGraph().to_dictionary(multiple_edges=True)
{0: [1, 4, 5], 1: [0, 2, 6],
 2: [1, 3, 7], 3: [2, 4, 8],
 4: [0, 3, 9], 5: [0, 7, 8],
 6: [1, 8, 9], 7: [2, 5, 9],
 8: [3, 5, 6], 9: [4, 6, 7]}
sage: graphs.PetersenGraph().to_dictionary(edge_labels=True)
{0: {1: None, 4: None, 5: None},
 1: {0: None, 2: None, 6: None},
 2: {1: None, 3: None, 7: None},
 3: {2: None, 4: None, 8: None},
 4: {0: None, 3: None, 9: None},
 5: {0: None, 7: None, 8: None},
 6: {1: None, 8: None, 9: None},
 7: {2: None, 5: None, 9: None},
 8: {3: None, 5: None, 6: None},
 9: {4: None, 6: None, 7: None}}
sage: graphs.PetersenGraph().to_dictionary(edge_labels=True,multiple_edges=True)
{0: {1: [None], 4: [None], 5: [None]},
 1: {0: [None], 2: [None], 6: [None]},
 2: {1: [None], 3: [None], 7: [None]},
 3: {2: [None], 4: [None], 8: [None]},
 4: {0: [None], 3: [None], 9: [None]},
 5: {0: [None], 7: [None], 8: [None]},
 6: {1: [None], 8: [None], 9: [None]},
 7: {2: [None], 5: [None], 9: [None]},
 8: {3: [None], 5: [None], 6: [None]},
 9: {4: [None], 6: [None], 7: [None]}}

to_simple(to_undirected=True, keep_label='any', immutable=None)

Returns a simple version of itself.

In particular, loops and multiple edges are removed, and the graph might optionally be converted to an undirected graph.

INPUT:

• to_undirected - boolean - if True, the graph is also converted to an undirected graph.
• keep_label ('any','min','max'): if there are multiple edges with different labels, this variable defines which label should be kept: any label ('any'), the smallest label ('min'), or the largest ('max').
• immutable (boolean) – whether to create a mutable/immutable copy. immutable=None (default) means that the graph and its copy will behave the same way.

EXAMPLES:

sage: G = DiGraph(loops=True,multiedges=True,sparse=True)
sage: G.add_edges( [ (0,0,None), (1,1,None), (2,2,None), (2,3,1), (2,3,2), (3,2,None) ] )
sage: G.edges(labels=False)
[(0, 0), (1, 1), (2, 2), (2, 3), (2, 3), (3, 2)]
sage: H=G.to_simple()
sage: H.edges(labels=False)
[(2, 3)]
sage: H.is_directed()
False
sage: H.allows_loops()
False
sage: H.allows_multiple_edges()
False
sage: G.to_simple(to_undirected=False, keep_label='min').edges()
[(2, 3, 1), (3, 2, None)]
sage: G.to_simple(to_undirected=False, keep_label='max').edges()
[(2, 3, 2), (3, 2, None)]

transitive_closure()

Computes the transitive closure of a graph and returns it. The original graph is not modified.

The transitive closure of a graph G has an edge (x,y) if and only if there is a path between x and y in G.

The transitive closure of any strongly connected component of a graph is a complete graph. In particular, the transitive closure of a connected undirected graph is a complete graph. The transitive closure of a directed acyclic graph is a directed acyclic graph representing the full partial order.

EXAMPLES:

sage: g=graphs.PathGraph(4)
sage: g.transitive_closure()
Transitive closure of Path graph: Graph on 4 vertices
sage: g.transitive_closure()==graphs.CompleteGraph(4)
True
sage: g=DiGraph({0:[1,2], 1:[3], 2:[4,5]})
sage: g.transitive_closure().edges(labels=False)
[(0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 3), (2, 4), (2, 5)]

On an immutable digraph:

sage: digraphs.Path(5).copy(immutable=True).transitive_closure()
Transitive closure of Path: Digraph on 5 vertices

transitive_reduction()

Returns a transitive reduction of a graph. The original graph is not modified.

A transitive reduction H of G has a path from x to y if and only if there was a path from x to y in G. Deleting any edge of H destroys this property. A transitive reduction is not unique in general. A transitive reduction has the same transitive closure as the original graph.

A transitive reduction of a complete graph is a tree. A transitive reduction of a tree is itself.

EXAMPLES:

sage: g = graphs.PathGraph(4)
sage: g.transitive_reduction() == g
True
sage: g = graphs.CompleteGraph(5)
sage: edges = g.transitive_reduction().edges(); len(edges)
4
sage: g = DiGraph({0:[1,2], 1:[2,3,4,5], 2:[4,5]})
sage: g.transitive_reduction().size()
5

traveling_salesman_problem(use_edge_labels=False, solver=None, constraint_generation=None, verbose=0, verbose_constraints=False)

Solves the traveling salesman problem (TSP)

Given a graph (resp. a digraph) \(G\) with weighted edges, the traveling salesman problem consists in finding a Hamiltonian cycle (resp. circuit) of the graph of minimum cost.

This TSP is one of the most famous NP-Complete problems, this function can thus be expected to take some time before returning its result.

INPUT:

• use_edge_labels (boolean) – whether to consider the weights of the edges.

  • If set to False (default), all edges are assumed to weight \(1\)
  • If set to True, the weights are taken into account, and the circuit returned is the one minimizing the sum of the weights (an edge with no label is assumed to have weight \(1\)).

• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• constraint_generation (boolean) – whether to use constraint generation when solving the Mixed Integer Linear Program.

  When constraint_generation = None, constraint generation is used whenever the graph has a density larger than 70%.

• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

• verbose_constraints – whether to display which constraints are being generated.

OUTPUT:

A solution to the TSP, as a Graph object whose vertex set is \(V(G)\), and whose edges are only those of the solution.

ALGORITHM:

This optimization problem is solved through the use of Linear Programming.

Note

This function is correctly defined for both graph and digraphs. In the second case, the returned cycle is a circuit of optimal cost.

EXAMPLES:

The Heawood graph is known to be Hamiltonian:

sage: g = graphs.HeawoodGraph()
sage: tsp = g.traveling_salesman_problem()
sage: tsp
TSP from Heawood graph: Graph on 14 vertices

The solution to the TSP has to be connected

sage: tsp.is_connected()
True

It must also be a \(2\)-regular graph:

sage: tsp.is_regular(k=2)
True

And obviously it is a subgraph of the Heawood graph:

sage: tsp.is_subgraph(g, induced=False)
True

On the other hand, the Petersen Graph is known not to be Hamiltonian:

sage: g = graphs.PetersenGraph()
sage: tsp = g.traveling_salesman_problem()
Traceback (most recent call last):
...
EmptySetError: The given graph is not Hamiltonian

One easy way to change it is obviously to add to this graph the edges corresponding to a Hamiltonian cycle. If we do this by setting the cost of these new edges to \(2\), while the others are set to \(1\), we notice that not all the edges we added are used in the optimal solution

sage: for u, v in g.edges(labels = None):
....:    g.set_edge_label(u,v,1)

sage: cycle = graphs.CycleGraph(10)
sage: for u,v in cycle.edges(labels = None):
....:    if not g.has_edge(u,v):
....:        g.add_edge(u,v)
....:    g.set_edge_label(u,v,2)

sage: tsp = g.traveling_salesman_problem(use_edge_labels = True)
sage: sum( tsp.edge_labels() ) < 2*10
True

If we pick \(1/2\) instead of \(2\) as a cost for these new edges, they clearly become the optimal solution:

sage: for u,v in cycle.edges(labels = None):
....:    g.set_edge_label(u,v,1/2)

sage: tsp = g.traveling_salesman_problem(use_edge_labels = True)
sage: sum( tsp.edge_labels() ) == (1/2)*10
True

triangles_count(algorithm=None)

Returns the number of triangles in the (di)graph.

For digraphs, we count the number of directed circuit of length 3.

INPUT:

• algorithm – (default: None) specifies the algorithm to use (note that only 'iter' is available for directed graphs):

  • 'sparse_copy' – counts the triangles in a sparse copy of the graph (see sage.graphs.base.static_sparse_graph). Calls static_sparse_graph.triangles_count
  • 'dense_copy' – counts the triangles in a dense copy of the graph (see sage.graphs.base.static_dense_graph). Calls static_dense_graph.triangles_count
  • 'matrix' uses the trace of the cube of the adjacency matrix.
  • 'iter' iterates over the pairs of neighbors of each vertex. No copy of the graph is performed
  • None – for undirected graphs, uses "sparse_copy" or "dense_copy" depending on whether the graph is stored as dense or sparse. For directed graphs, uses 'iter'.

EXAMPLES:

The Petersen graph is triangle free and thus:

sage: G = graphs.PetersenGraph()
sage: G.triangles_count()
0

Any triple of vertices in the complete graph induces a triangle so we have:

sage: G = graphs.CompleteGraph(150)
sage: G.triangles_count() == binomial(150,3)
True

The 2-dimensional DeBruijn graph of 2 symbols has 2 directed C3:

sage: G = digraphs.DeBruijn(2,2)
sage: G.triangles_count()
2

The directed n-cycle is trivially triangle free for n > 3:

sage: G = digraphs.Circuit(10)
sage: G.triangles_count()
0

union(other, immutable=None)

Returns the union of self and other.

If the graphs have common vertices, the common vertices will be identified.

If one of the two graphs allows loops (or multiple edges), the resulting graph will allow loops (or multiple edges).

If both graphs are immutable, the resulting graph is immutable, unless requested otherwise.

INPUT:

• immutable (boolean) – whether to create a mutable/immutable union. immutable=None (default) means that the graphs and their union will behave the same way.

See also

• disjoint_union()
• join()

EXAMPLES:

sage: G = graphs.CycleGraph(3)
sage: H = graphs.CycleGraph(4)
sage: J = G.union(H); J
Graph on 4 vertices
sage: J.vertices()
[0, 1, 2, 3]
sage: J.edges(labels=False)
[(0, 1), (0, 2), (0, 3), (1, 2), (2, 3)]

vertex_boundary(vertices1, vertices2=None)

Returns a list of all vertices in the external boundary of vertices1, intersected with vertices2. If vertices2 is None, then vertices2 is the complement of vertices1. This is much faster if vertices1 is smaller than vertices2.

The external boundary of a set of vertices is the union of the neighborhoods of each vertex in the set. Note that in this implementation, since vertices2 defaults to the complement of vertices1, if a vertex \(v\) has a loop, then vertex_boundary(v) will not contain \(v\).

In a digraph, the external boundary of a vertex v are those vertices u with an arc (v, u).

EXAMPLES:

sage: G = graphs.CubeGraph(4)
sage: l = ['0111', '0000', '0001', '0011', '0010', '0101', '0100', '1111', '1101', '1011', '1001']
sage: G.vertex_boundary(['0000', '1111'], l)
['0111', '0001', '0010', '0100', '1101', '1011']

sage: D = DiGraph({0:[1,2], 3:[0]})
sage: D.vertex_boundary([0])
[1, 2]

vertex_connectivity(value_only=True, sets=False, solver=None, verbose=0)

Returns the vertex connectivity of the graph. For more information, see the Wikipedia article on connectivity.

Note

• When the graph is a directed graph, this method actually computes the strong connectivity, (i.e. a directed graph is strongly \(k\)-connected if there are \(k\) disjoint paths between any two vertices \(u, v\)). If you do not want to consider strong connectivity, the best is probably to convert your DiGraph object to a Graph object, and compute the connectivity of this other graph.
• By convention, a complete graph on \(n\) vertices is \(n-1\) connected. In this case, no certificate can be given as there is no pair of vertices split by a cut of size \(k-1\). For this reason, the certificates returned in this situation are empty.

INPUT:

• value_only – boolean (default: True)
  • When set to True (default), only the value is returned.
  • When set to False , both the value and a minimum vertex cut are returned.
• sets – boolean (default: False)
  • When set to True, also returns the two sets of vertices that are disconnected by the cut. Implies value_only=False
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

EXAMPLES:

A basic application on a PappusGraph:

sage: g=graphs.PappusGraph()
sage: g.vertex_connectivity()
3

In a grid, the vertex connectivity is equal to the minimum degree, in which case one of the two sets is of cardinality \(1\):

sage: g = graphs.GridGraph([ 3,3 ])
sage: [value, cut, [ setA, setB ]] = g.vertex_connectivity(sets=True)
sage: len(setA) == 1 or len(setB) == 1
True

A vertex cut in a tree is any internal vertex:

sage: g = graphs.RandomGNP(15,.5)
sage: tree = Graph()
sage: tree.add_edges(g.min_spanning_tree())
sage: [val, [cut_vertex]] = tree.vertex_connectivity(value_only=False)
sage: tree.degree(cut_vertex) > 1
True

When value_only = True, this function is optimized for small connectivity values and does not need to build a linear program.

It is the case for connected graphs which are not connected:

sage: g = 2 * graphs.PetersenGraph()
sage: g.vertex_connectivity()
0

Or if they are just 1-connected:

sage: g = graphs.PathGraph(10)
sage: g.vertex_connectivity()
1

For directed graphs, the strong connectivity is tested through the dedicated function:

sage: g = digraphs.ButterflyGraph(3)
sage: g.vertex_connectivity()
0

A complete graph on \(10\) vertices is \(9\)-connected:

sage: g = graphs.CompleteGraph(10)
sage: g.vertex_connectivity()
9

A complete digraph on \(10\) vertices is \(9\)-connected:

sage: g = DiGraph(graphs.CompleteGraph(10))
sage: g.vertex_connectivity()
9

vertex_cut(s, t, value_only=True, vertices=False, solver=None, verbose=0)

Returns a minimum vertex cut between non-adjacent vertices \(s\) and \(t\) represented by a list of vertices.

A vertex cut between two non-adjacent vertices is a set \(U\) of vertices of self such that the graph obtained by removing \(U\) from self is disconnected. For more information, see the Wikipedia article on cuts.

INPUT:

• value_only – boolean (default: True). When set to True, only the size of the minimum cut is returned.
• vertices – boolean (default: False). When set to True, also returns the two sets of vertices that are disconnected by the cut. Implies value_only set to False.
• solver – (default: None) Specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0). Sets the level of verbosity. Set to 0 by default, which means quiet.

OUTPUT:

Real number or tuple, depending on the given arguments (examples are given below).

EXAMPLES:

A basic application in the Pappus graph:

sage: g = graphs.PappusGraph()
sage: g.vertex_cut(1, 16, value_only=True)
3

In the bipartite complete graph \(K_{2,8}\), a cut between the two vertices in the size \(2\) part consists of the other \(8\) vertices:

sage: g = graphs.CompleteBipartiteGraph(2, 8)
sage: [value, vertices] = g.vertex_cut(0, 1, value_only=False)
sage: print(value)
8
sage: vertices == list(range(2,10))
True

Clearly, in this case the two sides of the cut are singletons

sage: [value, vertices, [set1, set2]] = g.vertex_cut(0,1, vertices=True)
sage: len(set1) == 1
True
sage: len(set2) == 1
True

vertex_disjoint_paths(s, t)

Return a list of vertex-disjoint paths between two vertices as given by Menger’s theorem.

The vertex version of Menger’s theorem asserts that the size of the minimum vertex cut between two vertices \(s\) and \(t\) (the minimum number of vertices whose removal disconnects \(s\) and \(t\)) is equal to the maximum number of pairwise vertex-independent paths from \(s\) to \(t\).

This function returns a list of such paths.

EXAMPLES:

In a complete bipartite graph

sage: g = graphs.CompleteBipartiteGraph(2,3)
sage: g.vertex_disjoint_paths(0,1)
[[0, 2, 1], [0, 3, 1], [0, 4, 1]]

vertex_iterator(vertices=None)

Returns an iterator over the given vertices.

Returns False if not given a vertex, sequence, iterator or None. None is equivalent to a list of every vertex. Note that for v in G syntax is allowed.

INPUT:

• vertices - iterated vertices are these intersected with the vertices of the (di)graph

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: for v in P.vertex_iterator():
....:     print(v)
0
1
2
...
8
9

sage: G = graphs.TetrahedralGraph()
sage: for i in G:
....:     print(i)
0
1
2
3

Note that since the intersection option is available, the vertex_iterator() function is sub-optimal, speed-wise, but note the following optimization:

sage: timeit V = P.vertices()                   # not tested
100000 loops, best of 3: 8.85 [micro]s per loop
sage: timeit V = list(P.vertex_iterator())      # not tested
100000 loops, best of 3: 5.74 [micro]s per loop
sage: timeit V = list(P._nxg.adj)    # not tested
100000 loops, best of 3: 3.45 [micro]s per loop

In other words, if you want a fast vertex iterator, call the dictionary directly.

vertices(key=None)

Return a list of the vertices.

INPUT:

• key - default: None - a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm.

OUTPUT:

The vertices of the list.

Warning

There is always an attempt to sort the list before returning the result. However, since any object may be a vertex, there is no guarantee that any two vertices will be comparable. With default objects for vertices (all integers), or when all the vertices are of the same simple type, then there should not be a problem with how the vertices will be sorted. However, if you need to guarantee a total order for the sort, use the key argument, as illustrated in the examples below.

EXAMPLES:

sage: P = graphs.PetersenGraph()
sage: P.vertices()
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

If you do not care about sorted output and you are concerned about the time taken to sort, consider the following alternatives. The moral is: if you want a fast vertex iterator, call the dictionary directly.

sage: timeit V = P.vertices()                     # not tested
100000 loops, best of 3: 8.85 [micro]s per loop
sage: timeit V = list(P.vertex_iterator())        # not tested
100000 loops, best of 3: 5.74 [micro]s per loop
sage: timeit V = list(P._nxg.adj)      # not tested
100000 loops, best of 3: 3.45 [micro]s per loop

We illustrate various ways to use a key to sort the list:

sage: H=graphs.HanoiTowerGraph(3,3,labels=False)
sage: H.vertices()
[0, 1, 2, 3, 4, ... 22, 23, 24, 25, 26]
sage: H.vertices(key=lambda x: -x)
[26, 25, 24, 23, 22, ... 4, 3, 2, 1, 0]

sage: G=graphs.HanoiTowerGraph(3,3)
sage: G.vertices()
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), ... (2, 2, 1), (2, 2, 2)]
sage: G.vertices(key = lambda x: (x[1], x[2], x[0]))
[(0, 0, 0), (1, 0, 0), (2, 0, 0), (0, 0, 1), ... (1, 2, 2), (2, 2, 2)]

The discriminant of a polynomial is a function that returns an integer. We build a graph whose vertices are polynomials, and use the discriminant function to provide an ordering. Note that since functions are first-class objects in Python, we can specify precisely the function from the Sage library that we wish to use as the key.

sage: t = polygen(QQ, 't')
sage: K = Graph({5*t:[t^2], t^2:[t^2+2], t^2+2:[4*t^2-6], 4*t^2-6:[5*t]})
sage: dsc = sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint.discriminant
sage: verts = K.vertices(key=dsc)
sage: verts
[t^2 + 2, t^2, 5*t, 4*t^2 - 6]
sage: [x.discriminant() for x in verts]
[-8, 0, 1, 96]

weighted(new=None)

Whether the (di)graph is to be considered as a weighted (di)graph.

INPUT:

• new (optional bool): If it is provided, then the weightedness flag is set accordingly. This is not allowed for immutable graphs.

Note

Changing the weightedness flag changes the ==-class of a graph and is thus not allowed for immutable graphs.

Edge weightings can still exist for (di)graphs G where G.weighted() is False.

EXAMPLES:

Here we have two graphs with different labels, but weighted() is False for both, so we just check for the presence of edges:

sage: G = Graph({0:{1:'a'}}, sparse=True)
sage: H = Graph({0:{1:'b'}}, sparse=True)
sage: G == H
True

Now one is weighted and the other is not, and thus the graphs are not equal:

sage: G.weighted(True)
sage: H.weighted()
False
sage: G == H
False

However, if both are weighted, then we finally compare ‘a’ to ‘b’:

sage: H.weighted(True)
sage: G == H
False

weighted_adjacency_matrix(sparse=True)

Returns the weighted adjacency matrix of the graph.

Each vertex is represented by its position in the list returned by the vertices() function.

EXAMPLES:

sage: G = Graph(sparse=True, weighted=True)
sage: G.add_edges([(0,1,1),(1,2,2),(0,2,3),(0,3,4)])
sage: M = G.weighted_adjacency_matrix(); M
[0 1 3 4]
[1 0 2 0]
[3 2 0 0]
[4 0 0 0]
sage: H = Graph(data=M, format='weighted_adjacency_matrix', sparse=True)
sage: H == G
True

The following doctest verifies that trac ticket #4888 is fixed:

sage: G = DiGraph({0:{}, 1:{0:1}, 2:{0:1}}, weighted = True,sparse=True)
sage: G.weighted_adjacency_matrix()
[0 0 0]
[1 0 0]
[1 0 0]

wiener_index(by_weight=False, algorithm=None, weight_function=None, check_weight=True)

Return the Wiener index of the graph.

The graph is expected to have no cycles of negative weight.

The Wiener index of a graph \(G\) is \(W(G) = \frac 1 2 \sum_{u,v\in G} d(u,v)\) where \(d(u,v)\) denotes the distance between vertices \(u\) and \(v\) (see [KRG96b]).

For more information on the input variables and more examples, we refer to shortest_paths() and shortest_path_all_pairs(), which have very similar input variables.

INPUT:

• by_weight (boolean) - if True, the edges in the graph are weighted; if False, all edges have weight 1.
• algorithm (string) - the algorithm to use:
  • For by_weight==False only:
    • 'BFS' - the computation is done through a BFS centered on each vertex successively.
    • 'Floyd-Warshall-Cython' - the Cython implementation of the Floyd-Warshall algorithm. Usually slower than 'BFS'.
  • For graphs without negative weights:
    • 'Dijkstra_Boost': the Dijkstra algorithm, implemented in Boost.
    • 'Dijkstra_NetworkX': the Dijkstra algorithm, implemented in NetworkX. Usually slower than 'Dijkstra_Boost'.
  • For graphs with negative weights:
    • 'Johnson_Boost': the Johnson algorithm, implemented in Boost.
    • 'Floyd-Warshall-Python' - the Python implementation of the Floyd-Warshall algorithm. Usually slower than 'Johnson_Boost'.
  • None (default): Sage chooses the best algorithm: 'BFS' for unweighted graphs, 'Dijkstra_Boost' if all weights are positive, 'Johnson_Boost', otherwise.
• weight_function (function) - a function that inputs an edge (u, v, l) and outputs its weight. If not None, by_weight is automatically set to True. If None and by_weight is True, we use the edge label l as a weight.
• check_weight (boolean) - if True, we check that the weight_function outputs a number for each edge.

EXAMPLES:

sage: G = Graph( { 0: {1: None}, 1: {2: None}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True)
sage: G.wiener_index()
15
sage: G.wiener_index(weight_function=lambda e:(e[2] if e[2] is not None else 1))
20
sage: G.wiener_index(weight_function=lambda e:(e[2] if e[2] is not None else 200))
820
sage: G.wiener_index(algorithm='BFS')
15
sage: G.wiener_index(algorithm='Floyd-Warshall-Cython')
15
sage: G.wiener_index(algorithm='Floyd-Warshall-Python')
15
sage: G.wiener_index(algorithm='Dijkstra_Boost')
15
sage: G.wiener_index(algorithm='Johnson_Boost')
15
sage: G.wiener_index(algorithm='Dijkstra_NetworkX')
15

sage.graphs.generic_graph.graph_isom_equivalent_non_edge_labeled_graph(g, partition=None, standard_label=None, return_relabeling=False, return_edge_labels=False, inplace=False, ignore_edge_labels=False)

Helper function for canonical labeling of edge labeled (di)graphs.

Translates to a bipartite incidence-structure type graph appropriate for computing canonical labels of edge labeled and/or multi-edge graphs. Note that this is actually computationally equivalent to implementing a change on an inner loop of the main algorithm- namely making the refinement procedure sort for each label.

If the graph is a multigraph, it is translated to a non-multigraph, where each edge is labeled with a dictionary describing how many edges of each label were originally there. Then in either case we are working on a graph without multiple edges. At this point, we create another (bipartite) graph, whose left vertices are the original vertices of the graph, and whose right vertices represent the edges. We partition the left vertices as they were originally, and the right vertices by common labels: only automorphisms taking edges to like-labeled edges are allowed, and this additional partition information enforces this on the bipartite graph.

INPUT:

• g – Graph or DiGraph
• partition – (default:None) if given, the partition of the vertices is as well relabeled
• standard_label – (default:None) the standard label is not considered to be changed
• return_relabeling – (default: False) if True, a dictionary containing the relabeling is returned
• return_edge_labels – (default: False) if True, the different edge_labels are returned (useful if inplace is True)
• inplace – (default:False) if True, g is modified, otherwise the result is returned. Note that attributes of g are not copied for speed issues, only edges and vertices.

OUTPUT:

• if not inplace: the unlabeled graph without multiple edges
• the partition of the vertices
• if return_relabeling: a dictionary containing the relabeling
• if return_edge_labels: the list of (former) edge labels is returned

EXAMPLES:

sage: from sage.graphs.generic_graph import graph_isom_equivalent_non_edge_labeled_graph

sage: G = Graph(multiedges=True,sparse=True)
sage: G.add_edges( (0,1,i) for i in range(10) )
sage: G.add_edge(1,2,'string')
sage: G.add_edge(2,123)
sage: g = graph_isom_equivalent_non_edge_labeled_graph(G, partition=[[0,123],[1,2]]); g
[Graph on 6 vertices, [[0, 3], [1, 2], [4], [5]]]

sage: g = graph_isom_equivalent_non_edge_labeled_graph(G); g
[Graph on 6 vertices, [[0, 1, 2, 3], [4], [5]]]
sage: g[0].edges()
[(0, 4, None), (1, 4, None), (1, 5, None), (2, 3, None), (2, 5, None)]

sage: g = graph_isom_equivalent_non_edge_labeled_graph(G,standard_label='string',return_edge_labels=True); g
[Graph on 6 vertices, [[0, 1, 2, 3], [5], [4]], [[[None, 1]], [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1]], [['string', 1]]]]
sage: g[0].edges()
[(0, 4, None), (1, 2, None), (1, 4, None), (2, 5, None), (3, 5, None)]

sage: graph_isom_equivalent_non_edge_labeled_graph(G,inplace=True)
[[[0, 1, 2, 3], [4], [5]]]
sage: G.edges()
[(0, 4, None), (1, 4, None), (1, 5, None), (2, 3, None), (2, 5, None)]

Ensure that trac ticket #14108 is fixed:

sage: G=DiGraph({0:[0,0,0],1:[1,1,1]})
sage: H=DiGraph({0:[0,0,0,0],1:[1,1]})
sage: G.is_isomorphic(H)
False
sage: H=DiGraph({0:[0,0,0,0],1:[1,1]})
sage: HH=DiGraph({0:[0,0,0],1:[1,1,1]})
sage: H.is_isomorphic(HH)
False
sage: H.is_isomorphic(HH, edge_labels=True)
False

sage.graphs.generic_graph.tachyon_vertex_plot(g, bgcolor=(1, 1, 1), vertex_colors=None, vertex_size=0.06, pos3d=None, **kwds)

Helper function for plotting graphs in 3d with Tachyon. Returns a plot containing only the vertices, as well as the 3d position dictionary used for the plot.

INPUT:

• \(pos3d\) - a 3D layout of the vertices
• various rendering options

EXAMPLES:

sage: G = graphs.TetrahedralGraph()
sage: from sage.graphs.generic_graph import tachyon_vertex_plot
sage: T,p = tachyon_vertex_plot(G, pos3d = G.layout(dim=3))
sage: type(T)
<class 'sage.plot.plot3d.tachyon.Tachyon'>
sage: type(p)
<... 'dict'>