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Definition df-crcts 26737
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 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-crcts
StepHypRef Expression
1 ccrcts 26735 . 2 class Circuits
2 vg . . 3 setvar 𝑔
3 cvv 3231 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1522 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1522 . . . . . 6 class 𝑝
82cv 1522 . . . . . . 7 class 𝑔
9 ctrls 26643 . . . . . . 7 class Trails
108, 9cfv 5926 . . . . . 6 class (Trails‘𝑔)
115, 7, 10wbr 4685 . . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12 cc0 9974 . . . . . . 7 class 0
1312, 7cfv 5926 . . . . . 6 class (𝑝‘0)
14 chash 13157 . . . . . . . 8 class #
155, 14cfv 5926 . . . . . . 7 class (#‘𝑓)
1615, 7cfv 5926 . . . . . 6 class (𝑝‘(#‘𝑓))
1713, 16wceq 1523 . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
1811, 17wa 383 . . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
1918, 4, 6copab 4745 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
202, 3, 19cmpt 4762 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
211, 20wceq 1523 1 wff Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
Colors of variables: wff setvar class
This definition is referenced by:  crcts  26739
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