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Mirrors > Home > MPE Home > Th. List > df-cycl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-cycl | ⊢ Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccycl 26035 | . 2 class Cycles | |
2 | vv | . . 3 setvar 𝑣 | |
3 | ve | . . 3 setvar 𝑒 | |
4 | cvv 3173 | . . 3 class V | |
5 | vf | . . . . . . 7 setvar 𝑓 | |
6 | 5 | cv 1474 | . . . . . 6 class 𝑓 |
7 | vp | . . . . . . 7 setvar 𝑝 | |
8 | 7 | cv 1474 | . . . . . 6 class 𝑝 |
9 | 2 | cv 1474 | . . . . . . 7 class 𝑣 |
10 | 3 | cv 1474 | . . . . . . 7 class 𝑒 |
11 | cpath 26028 | . . . . . . 7 class Paths | |
12 | 9, 10, 11 | co 6549 | . . . . . 6 class (𝑣 Paths 𝑒) |
13 | 6, 8, 12 | wbr 4583 | . . . . 5 wff 𝑓(𝑣 Paths 𝑒)𝑝 |
14 | cc0 9815 | . . . . . . 7 class 0 | |
15 | 14, 8 | cfv 5804 | . . . . . 6 class (𝑝‘0) |
16 | chash 12979 | . . . . . . . 8 class # | |
17 | 6, 16 | cfv 5804 | . . . . . . 7 class (#‘𝑓) |
18 | 17, 8 | cfv 5804 | . . . . . 6 class (𝑝‘(#‘𝑓)) |
19 | 15, 18 | wceq 1475 | . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓)) |
20 | 13, 19 | wa 383 | . . . 4 wff (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓))) |
21 | 20, 5, 7 | copab 4642 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} |
22 | 2, 3, 4, 4, 21 | cmpt2 6551 | . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
23 | 1, 22 | wceq 1475 | 1 wff Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
Colors of variables: wff setvar class |
This definition is referenced by: cycls 26151 cyclispth 26157 cycliscrct 26158 cyclnspth 26159 |
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