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Definition df-spth 25084
 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.)
Assertion
Ref Expression
df-spth SPaths Trails
Distinct variable group:   ,,,

Detailed syntax breakdown of Definition df-spth
StepHypRef Expression
1 cspath 25074 . 2 SPaths
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 ctrail 25072 . . . . . . 7 Trails
129, 10, 11co 6305 . . . . . 6 Trails
136, 8, 12wbr 4426 . . . . 5 Trails
148ccnv 4853 . . . . . 6
1514wfun 5595 . . . . 5
1613, 15wa 370 . . . 4 Trails
1716, 5, 7copab 4483 . . 3 Trails
182, 3, 4, 4, 17cmpt2 6307 . 2 Trails
191, 18wceq 1437 1 SPaths Trails
 Colors of variables: wff setvar class This definition is referenced by:  spths  25142  spthispth  25148
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