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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-upwlks | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-upwlks | ⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cupwlks 45936 | . 2 class UPWalks | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3443 | . . 3 class V | |
4 | vf | . . . . . . 7 setvar 𝑓 | |
5 | 4 | cv 1540 | . . . . . 6 class 𝑓 |
6 | 2 | cv 1540 | . . . . . . . . 9 class 𝑔 |
7 | ciedg 27777 | . . . . . . . . 9 class iEdg | |
8 | 6, 7 | cfv 6493 | . . . . . . . 8 class (iEdg‘𝑔) |
9 | 8 | cdm 5631 | . . . . . . 7 class dom (iEdg‘𝑔) |
10 | 9 | cword 14356 | . . . . . 6 class Word dom (iEdg‘𝑔) |
11 | 5, 10 | wcel 2106 | . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔) |
12 | cc0 11009 | . . . . . . 7 class 0 | |
13 | chash 14184 | . . . . . . . 8 class ♯ | |
14 | 5, 13 | cfv 6493 | . . . . . . 7 class (♯‘𝑓) |
15 | cfz 13378 | . . . . . . 7 class ... | |
16 | 12, 14, 15 | co 7351 | . . . . . 6 class (0...(♯‘𝑓)) |
17 | cvtx 27776 | . . . . . . 7 class Vtx | |
18 | 6, 17 | cfv 6493 | . . . . . 6 class (Vtx‘𝑔) |
19 | vp | . . . . . . 7 setvar 𝑝 | |
20 | 19 | cv 1540 | . . . . . 6 class 𝑝 |
21 | 16, 18, 20 | wf 6489 | . . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) |
22 | vk | . . . . . . . . . 10 setvar 𝑘 | |
23 | 22 | cv 1540 | . . . . . . . . 9 class 𝑘 |
24 | 23, 5 | cfv 6493 | . . . . . . . 8 class (𝑓‘𝑘) |
25 | 24, 8 | cfv 6493 | . . . . . . 7 class ((iEdg‘𝑔)‘(𝑓‘𝑘)) |
26 | 23, 20 | cfv 6493 | . . . . . . . 8 class (𝑝‘𝑘) |
27 | c1 11010 | . . . . . . . . . 10 class 1 | |
28 | caddc 11012 | . . . . . . . . . 10 class + | |
29 | 23, 27, 28 | co 7351 | . . . . . . . . 9 class (𝑘 + 1) |
30 | 29, 20 | cfv 6493 | . . . . . . . 8 class (𝑝‘(𝑘 + 1)) |
31 | 26, 30 | cpr 4586 | . . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} |
32 | 25, 31 | wceq 1541 | . . . . . 6 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} |
33 | cfzo 13521 | . . . . . . 7 class ..^ | |
34 | 12, 14, 33 | co 7351 | . . . . . 6 class (0..^(♯‘𝑓)) |
35 | 32, 22, 34 | wral 3062 | . . . . 5 wff ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} |
36 | 11, 21, 35 | w3a 1087 | . . . 4 wff (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) |
37 | 36, 4, 19 | copab 5165 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} |
38 | 2, 3, 37 | cmpt 5186 | . 2 class (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
39 | 1, 38 | wceq 1541 | 1 wff UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
Colors of variables: wff setvar class |
This definition is referenced by: upwlksfval 45938 |
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