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Definition df-clwlk 25470
 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 { 1 , ... , n } 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(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0). Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 25485! (Contributed by Alexander van der Vekens, 12-Mar-2018.)
Assertion
Ref Expression
df-clwlk ClWalks Walks
Distinct variable group:   ,,,

Detailed syntax breakdown of Definition df-clwlk
StepHypRef Expression
1 cclwlk 25467 . 2 ClWalks
2 vv . . 3
3 ve . . 3
4 cvv 3082 . . 3
5 vf . . . . . . 7
65cv 1437 . . . . . 6
7 vp . . . . . . 7
87cv 1437 . . . . . 6
92cv 1437 . . . . . . 7
103cv 1437 . . . . . . 7
11 cwalk 25218 . . . . . . 7 Walks
129, 10, 11co 6303 . . . . . 6 Walks
136, 8, 12wbr 4421 . . . . 5 Walks
14 cc0 9541 . . . . . . 7
1514, 8cfv 5599 . . . . . 6
16 chash 12516 . . . . . . . 8
176, 16cfv 5599 . . . . . . 7
1817, 8cfv 5599 . . . . . 6
1915, 18wceq 1438 . . . . 5
2013, 19wa 371 . . . 4 Walks
2120, 5, 7copab 4479 . . 3 Walks
222, 3, 4, 4, 21cmpt2 6305 . 2 Walks
231, 22wceq 1438 1 ClWalks Walks
 Colors of variables: wff setvar class This definition is referenced by:  clwlk  25473  isclwlkg  25475  clwlkiswlk  25477  clwlkswlks  25478  clwlkcompim  25484
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