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Mirrors > Home > MPE Home > Th. List > df-clwlks | Structured version Visualization version Unicode version |
Description: Define the set of all
closed walks (in an undirected graph).
According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0". According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0). Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 26991! (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.) |
Ref | Expression |
---|---|
df-clwlks | ClWalks Walks |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cclwlks 26666 | . 2 ClWalks | |
2 | vg | . . 3 | |
3 | cvv 3200 | . . 3 | |
4 | vf | . . . . . . 7 | |
5 | 4 | cv 1482 | . . . . . 6 |
6 | vp | . . . . . . 7 | |
7 | 6 | cv 1482 | . . . . . 6 |
8 | 2 | cv 1482 | . . . . . . 7 |
9 | cwlks 26492 | . . . . . . 7 Walks | |
10 | 8, 9 | cfv 5888 | . . . . . 6 Walks |
11 | 5, 7, 10 | wbr 4653 | . . . . 5 Walks |
12 | cc0 9936 | . . . . . . 7 | |
13 | 12, 7 | cfv 5888 | . . . . . 6 |
14 | chash 13117 | . . . . . . . 8 | |
15 | 5, 14 | cfv 5888 | . . . . . . 7 |
16 | 15, 7 | cfv 5888 | . . . . . 6 |
17 | 13, 16 | wceq 1483 | . . . . 5 |
18 | 11, 17 | wa 384 | . . . 4 Walks |
19 | 18, 4, 6 | copab 4712 | . . 3 Walks |
20 | 2, 3, 19 | cmpt 4729 | . 2 Walks |
21 | 1, 20 | wceq 1483 | 1 ClWalks Walks |
Colors of variables: wff setvar class |
This definition is referenced by: clwlks 26668 |
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