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Definition df-cycls 26914
Description: Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... 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-cycls Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-cycls
StepHypRef Expression
1 ccycls 26912 . 2 class Cycles
2 vg . . 3 setvar 𝑔
3 cvv 3340 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1631 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1631 . . . . . 6 class 𝑝
82cv 1631 . . . . . . 7 class 𝑔
9 cpths 26839 . . . . . . 7 class Paths
108, 9cfv 6049 . . . . . 6 class (Paths‘𝑔)
115, 7, 10wbr 4804 . . . . 5 wff 𝑓(Paths‘𝑔)𝑝
12 cc0 10148 . . . . . . 7 class 0
1312, 7cfv 6049 . . . . . 6 class (𝑝‘0)
14 chash 13331 . . . . . . . 8 class
155, 14cfv 6049 . . . . . . 7 class (♯‘𝑓)
1615, 7cfv 6049 . . . . . 6 class (𝑝‘(♯‘𝑓))
1713, 16wceq 1632 . . . . 5 wff (𝑝‘0) = (𝑝‘(♯‘𝑓))
1811, 17wa 383 . . . 4 wff (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))
1918, 4, 6copab 4864 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
202, 3, 19cmpt 4881 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
211, 20wceq 1632 1 wff Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
Colors of variables: wff setvar class
This definition is referenced by:  cycls  26916
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