Metamath Proof Explorer


Definition df-trls

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) (Revised by AV, 28-Dec-2020)

Ref Expression
Assertion df-trls Trails = g V f p | f Walks g p Fun f -1

Detailed syntax breakdown

Step Hyp Ref Expression
0 ctrls class Trails
1 vg setvar g
2 cvv class V
3 vf setvar f
4 vp setvar p
5 3 cv setvar f
6 cwlks class Walks
7 1 cv setvar g
8 7 6 cfv class Walks g
9 4 cv setvar p
10 5 9 8 wbr wff f Walks g p
11 5 ccnv class f -1
12 11 wfun wff Fun f -1
13 10 12 wa wff f Walks g p Fun f -1
14 13 3 4 copab class f p | f Walks g p Fun f -1
15 1 2 14 cmpt class g V f p | f Walks g p Fun f -1
16 0 15 wceq wff Trails = g V f p | f Walks g p Fun f -1