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Definition df-pth 25083
 Description: Define the set of all 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." According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 25183). Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the 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 and Mario Carneiro, 4-Oct-2017.)
Assertion
Ref Expression
df-pth Paths Trails ..^ ..^
Distinct variable group:   ,,,

Detailed syntax breakdown of Definition df-pth
StepHypRef Expression
1 cpath 25073 . 2 Paths
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
14 c1 9539 . . . . . . . . 9
15 chash 12512 . . . . . . . . . 10
166, 15cfv 5601 . . . . . . . . 9
17 cfzo 11913 . . . . . . . . 9 ..^
1814, 16, 17co 6305 . . . . . . . 8 ..^
198, 18cres 4856 . . . . . . 7 ..^
2019ccnv 4853 . . . . . 6 ..^
2120wfun 5595 . . . . 5 ..^
22 cc0 9538 . . . . . . . . 9
2322, 16cpr 4004 . . . . . . . 8
248, 23cima 4857 . . . . . . 7
258, 18cima 4857 . . . . . . 7 ..^
2624, 25cin 3441 . . . . . 6 ..^
27 c0 3767 . . . . . 6
2826, 27wceq 1437 . . . . 5 ..^
2913, 21, 28w3a 982 . . . 4 Trails ..^ ..^
3029, 5, 7copab 4483 . . 3 Trails ..^ ..^
312, 3, 4, 4, 30cmpt2 6307 . 2 Trails ..^ ..^
321, 31wceq 1437 1 Paths Trails ..^ ..^
 Colors of variables: wff setvar class This definition is referenced by:  pths  25141  pthistrl  25147  pthdepisspth  25149
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