Description: Define the set of all simple paths (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory) , 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."
Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path 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, 20-Oct-2017) (Revised by AV, 9-Jan-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | df-spths |
Step | Hyp | Ref | Expression |
---|---|---|---|
0 | cspths | ||
1 | vg | ||
2 | cvv | ||
3 | vf | ||
4 | vp | ||
5 | 3 | cv | |
6 | ctrls | ||
7 | 1 | cv | |
8 | 7 6 | cfv | |
9 | 4 | cv | |
10 | 5 9 8 | wbr | |
11 | 9 | ccnv | |
12 | 11 | wfun | |
13 | 10 12 | wa | |
14 | 13 3 4 | copab | |
15 | 1 2 14 | cmpt | |
16 | 0 15 | wceq |