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 44349
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 pseudograhs 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 44348 . 2 class UPWalks
2 vg . . 3 setvar 𝑔
3 cvv 3444 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1537 . . . . . 6 class 𝑓
62cv 1537 . . . . . . . . 9 class 𝑔
7 ciedg 26793 . . . . . . . . 9 class iEdg
86, 7cfv 6328 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5523 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 13861 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 2112 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 10530 . . . . . . 7 class 0
13 chash 13690 . . . . . . . 8 class
145, 13cfv 6328 . . . . . . 7 class (♯‘𝑓)
15 cfz 12889 . . . . . . 7 class ...
1612, 14, 15co 7139 . . . . . 6 class (0...(♯‘𝑓))
17 cvtx 26792 . . . . . . 7 class Vtx
186, 17cfv 6328 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1537 . . . . . 6 class 𝑝
2116, 18, 20wf 6324 . . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1537 . . . . . . . . 9 class 𝑘
2423, 5cfv 6328 . . . . . . . 8 class (𝑓𝑘)
2524, 8cfv 6328 . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓𝑘))
2623, 20cfv 6328 . . . . . . . 8 class (𝑝𝑘)
27 c1 10531 . . . . . . . . . 10 class 1
28 caddc 10533 . . . . . . . . . 10 class +
2923, 27, 28co 7139 . . . . . . . . 9 class (𝑘 + 1)
3029, 20cfv 6328 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
3126, 30cpr 4530 . . . . . . 7 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3225, 31wceq 1538 . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
33 cfzo 13032 . . . . . . 7 class ..^
3412, 14, 33co 7139 . . . . . 6 class (0..^(♯‘𝑓))
3532, 22, 34wral 3109 . . . . 5 wff 𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3611, 21, 35w3a 1084 . . . 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 5095 . . 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 5113 . 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 1538 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  44350
  Copyright terms: Public domain W3C validator