Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-upwlks Structured version   Visualization version   GIF version

Definition df-upwlks 46122
Description: Define the set of all walks (in a pseudograph), called "simple walks" in the following.

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n).

Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

Assertion
Ref Expression
df-upwlks UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
Distinct variable group:   𝑓,𝑔,𝑘,𝑝

Detailed syntax breakdown of Definition df-upwlks
StepHypRef Expression
1 cupwlks 46121 . 2 class UPWalks
2 vg . . 3 setvar 𝑔
3 cvv 3444 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1541 . . . . . 6 class 𝑓
62cv 1541 . . . . . . . . 9 class 𝑔
7 ciedg 27990 . . . . . . . . 9 class iEdg
86, 7cfv 6497 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5634 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 14408 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 2107 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 11056 . . . . . . 7 class 0
13 chash 14236 . . . . . . . 8 class
145, 13cfv 6497 . . . . . . 7 class (♯‘𝑓)
15 cfz 13430 . . . . . . 7 class ...
1612, 14, 15co 7358 . . . . . 6 class (0...(♯‘𝑓))
17 cvtx 27989 . . . . . . 7 class Vtx
186, 17cfv 6497 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1541 . . . . . 6 class 𝑝
2116, 18, 20wf 6493 . . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1541 . . . . . . . . 9 class 𝑘
2423, 5cfv 6497 . . . . . . . 8 class (𝑓𝑘)
2524, 8cfv 6497 . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓𝑘))
2623, 20cfv 6497 . . . . . . . 8 class (𝑝𝑘)
27 c1 11057 . . . . . . . . . 10 class 1
28 caddc 11059 . . . . . . . . . 10 class +
2923, 27, 28co 7358 . . . . . . . . 9 class (𝑘 + 1)
3029, 20cfv 6497 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
3126, 30cpr 4589 . . . . . . 7 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3225, 31wceq 1542 . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
33 cfzo 13573 . . . . . . 7 class ..^
3412, 14, 33co 7358 . . . . . 6 class (0..^(♯‘𝑓))
3532, 22, 34wral 3061 . . . . 5 wff 𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3611, 21, 35w3a 1088 . . . 4 wff (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})
3736, 4, 19copab 5168 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})}
382, 3, 37cmpt 5189 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
391, 38wceq 1542 1 wff UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
Colors of variables: wff setvar class
This definition is referenced by:  upwlksfval  46123
  Copyright terms: Public domain W3C validator