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Definition df-clwwlk 25321
 Description: Define the set of all Closed Walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlk 25320. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk. Notice that by this definition, a single vertex cannot be represented as closed walk, since the word <" v "> with vertex v represents the walk "vv", which is a (closed) walk of length 1 (if there is an edge/loop from v to v). Therefore, a closed walk corresponds to a closed walk as word in an undirected graph only for walks of length at least 1, see clwlkisclwwlk2 25360. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
df-clwwlk ClWWalks Word ..^ lastS
Distinct variable group:   ,,,

Detailed syntax breakdown of Definition df-clwwlk
StepHypRef Expression
1 cclwwlk 25318 . 2 ClWWalks
2 vv . . 3
3 ve . . 3
4 cvv 3078 . . 3
5 vi . . . . . . . . . 10
65cv 1436 . . . . . . . . 9
7 vw . . . . . . . . . 10
87cv 1436 . . . . . . . . 9
96, 8cfv 5592 . . . . . . . 8
10 c1 9529 . . . . . . . . . 10
11 caddc 9531 . . . . . . . . . 10
126, 10, 11co 6296 . . . . . . . . 9
1312, 8cfv 5592 . . . . . . . 8
149, 13cpr 3995 . . . . . . 7
153cv 1436 . . . . . . . 8
1615crn 4846 . . . . . . 7
1714, 16wcel 1867 . . . . . 6
18 cc0 9528 . . . . . . 7
19 chash 12501 . . . . . . . . 9
208, 19cfv 5592 . . . . . . . 8
21 cmin 9849 . . . . . . . 8
2220, 10, 21co 6296 . . . . . . 7
23 cfzo 11902 . . . . . . 7 ..^
2418, 22, 23co 6296 . . . . . 6 ..^
2517, 5, 24wral 2773 . . . . 5 ..^
26 clsw 12633 . . . . . . . 8 lastS
278, 26cfv 5592 . . . . . . 7 lastS
2818, 8cfv 5592 . . . . . . 7
2927, 28cpr 3995 . . . . . 6 lastS
3029, 16wcel 1867 . . . . 5 lastS
3125, 30wa 370 . . . 4 ..^ lastS
322cv 1436 . . . . 5
3332cword 12632 . . . 4 Word
3431, 7, 33crab 2777 . . 3 Word ..^ lastS
352, 3, 4, 4, 34cmpt2 6298 . 2 Word ..^ lastS
361, 35wceq 1437 1 ClWWalks Word ..^ lastS
 Colors of variables: wff setvar class This definition is referenced by:  clwwlk  25336  clwwlkprop  25340
 Copyright terms: Public domain W3C validator