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Definition df-trail 25082
 Description: Define the set of all Trails (in an undirected graph). According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct. According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5. Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail 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) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
Assertion
Ref Expression
df-trail Trails Walks
Distinct variable group:   ,,,

Detailed syntax breakdown of Definition df-trail
StepHypRef Expression
1 ctrail 25072 . 2 Trails
2 vv . . 3
3 ve . . 3
4 cvv 3087 . . 3
5 vf . . . . . . 7
65cv 1436 . . . . . 6
7 vp . . . . . . 7
87cv 1436 . . . . . 6
92cv 1436 . . . . . . 7
103cv 1436 . . . . . . 7
11 cwalk 25071 . . . . . . 7 Walks
129, 10, 11co 6305 . . . . . 6 Walks
136, 8, 12wbr 4426 . . . . 5 Walks
146ccnv 4853 . . . . . 6
1514wfun 5595 . . . . 5
1613, 15wa 370 . . . 4 Walks
1716, 5, 7copab 4483 . . 3 Walks
182, 3, 4, 4, 17cmpt2 6307 . 2 Walks
191, 18wceq 1437 1 Trails Walks
 Colors of variables: wff setvar class This definition is referenced by:  trls  25111  trliswlk  25114
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