Definition df-crcts 26681

Description: Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)

Assertion

Ref	Expression
df-crcts	|- Circuits = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )

Distinct variable group:   f, g, p

Detailed syntax breakdown of Definition df-crcts

Step	Hyp	Ref	Expression
1		ccrcts 26679	. 2 class Circuits
2		vg	. . 3 setvar g
3		cvv 3200	. . 3 class _V
4		vf	. . . . . . 7 setvar f
5	4	cv 1482	. . . . . 6 class f
6		vp	. . . . . . 7 setvar p
7	6	cv 1482	. . . . . 6 class p
8	2	cv 1482	. . . . . . 7 class g
9		ctrls 26587	. . . . . . 7 class Trails
10	8, 9	cfv 5888	. . . . . 6 class (Trails ` g )
11	5, 7, 10	wbr 4653	. . . . 5 wff f (Trails ` g ) p
12		cc0 9936	. . . . . . 7 class 0
13	12, 7	cfv 5888	. . . . . 6 class ( p ` 0 )
14		chash 13117	. . . . . . . 8 class #
15	5, 14	cfv 5888	. . . . . . 7 class ( # ` f )
16	15, 7	cfv 5888	. . . . . 6 class ( p ` ( # ` f ) )
17	13, 16	wceq 1483	. . . . 5 wff ( p ` 0 ) = ( p ` ( # ` f ) )
18	11, 17	wa 384	. . . 4 wff ( f (Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) )
19	18, 4, 6	copab 4712	. . 3 class { <. f , p >. | ( f (Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }
20	2, 3, 19	cmpt 4729	. 2 class ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
21	1, 20	wceq 1483	1 wff Circuits = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )

This definition is referenced by:  crcts  26683