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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisspth 26801 Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
(𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))

Theorempthistrl 26802 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Paths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)

Theoremspthispth 26803 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(SPaths‘𝐺)𝑃𝐹(Paths‘𝐺)𝑃)

Theorempthiswlk 26804 A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
(𝐹(Paths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremspthiswlk 26805 A simple path is a walk (in an undirected graph). (Contributed by AV, 16-May-2021.)
(𝐹(SPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theorempthdivtx 26806 The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼𝐽)) → (𝑃𝐼) ≠ (𝑃𝐽))

Theorempthdadjvtx 26807 The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 𝐼 ∈ (0..^(♯‘𝐹))) → (𝑃𝐼) ≠ (𝑃‘(𝐼 + 1)))

Theorem2pthnloop 26808* A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 27274. (Contributed by AV, 6-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹𝑖))))

Theoremupgr2pthnlp 26809* A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))(♯‘(𝐼‘(𝐹𝑖))) = 2)

Theoremspthdifv 26810 The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(SPaths‘𝐺)𝑃𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺))

Theoremspthdep 26811 A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
((𝐹(SPaths‘𝐺)𝑃 ∧ (♯‘𝐹) ≠ 0) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))

Theorempthdepisspth 26812 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝐹(SPaths‘𝐺)𝑃)

Theoremupgrwlkdvdelem 26813* Lemma for upgrwlkdvde 26814. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)
((𝑃:(0...(♯‘𝐹))–1-1𝑉𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹))

Theoremupgrwlkdvde 26814 In a pseudograph, all edges of a walk consisting of different vertices are different. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 26815. (Contributed by AV, 17-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → Fun 𝐹)

Theoremupgrspthswlk 26815* The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})

Theoremupgrwlkdvspth 26816 A walk consisting of different vertices is a simple path. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 26815. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → 𝐹(SPaths‘𝐺)𝑃)

Theorempthsonfval 26817* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)})

Theoremspthson 26818* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(SPathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(SPaths‘𝐺)𝑝)})

Theoremispthson 26819 Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))

Theoremisspthson 26820 Properties of a pair of functions to be a simple path between two given vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)))

Theorempthsonprop 26821 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))

Theoremspthonprop 26822 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)))

Theorempthonispth 26823 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 17-Jan-2021.)
(𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)

Theorempthontrlon 26824 A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.)
(𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃)

Theorempthonpth 26825 A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.)
(𝐹(Paths‘𝐺)𝑃𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)

Theoremisspthonpth 26826 A pair of functions is a simple path between two given vertices iff it is a simple path starting and ending at the two vertices. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))

Theoremspthonisspth 26827 A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 18-Jan-2021.)
(𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)

Theoremspthonpthon 26828 A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021.)
(𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃)

Theoremspthonepeq 26829 The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
(𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))

Theoremuhgrwkspthlem1 26830 Lemma 1 for uhgrwkspth 26832. (Contributed by AV, 25-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 1) → Fun 𝐹)

Theoremuhgrwkspthlem2 26831 Lemma 2 for uhgrwkspth 26832. (Contributed by AV, 25-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ ((♯‘𝐹) = 1 ∧ 𝐴𝐵) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) → Fun 𝑃)

Theoremuhgrwkspth 26832 Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺𝑊 ∧ (♯‘𝐹) = 1 ∧ 𝐴𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))

Theoremusgr2wlkneq 26833 The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
(((𝐺 ∈ USGraph ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ∧ (𝐹‘0) ≠ (𝐹‘1)))

Theoremusgr2wlkspthlem1 26834 Lemma 1 for usgr2wlkspth 26836. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun 𝐹)

Theoremusgr2wlkspthlem2 26835 Lemma 2 for usgr2wlkspth 26836. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun 𝑃)

Theoremusgr2wlkspth 26836 In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ 𝐴𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))

Theoremusgr2trlncl 26837 In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))

Theoremusgr2trlspth 26838 In a simple graph, any trail of length 2 is a simple path. (Contributed by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃𝐹(SPaths‘𝐺)𝑃))

Theoremusgr2pthspth 26839 In a simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.) (Revised by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃𝐹(SPaths‘𝐺)𝑃))

Theoremusgr2pthlem 26840* Lemma for usgr2pth 26841. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))

Theoremusgr2pth 26841* In a simple graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))

Theoremusgr2pth0 26842* In a simply graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼𝑃:(0...2)–1-1𝑉 ∧ ∃𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))

Theorempthdlem1 26843* Lemma 1 for pthd 26846. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   𝑅 = ((♯‘𝑃) − 1)    &   (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖𝑗 → (𝑃𝑖) ≠ (𝑃𝑗)))       (𝜑 → Fun (𝑃 ↾ (1..^𝑅)))

Theorempthdlem2lem 26844* Lemma for pthdlem2 26845. (Contributed by AV, 10-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   𝑅 = ((♯‘𝑃) − 1)    &   (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖𝑗 → (𝑃𝑖) ≠ (𝑃𝑗)))       ((𝜑 ∧ (♯‘𝑃) ∈ ℕ ∧ (𝐼 = 0 ∨ 𝐼 = 𝑅)) → (𝑃𝐼) ∉ (𝑃 “ (1..^𝑅)))

Theorempthdlem2 26845* Lemma 2 for pthd 26846. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   𝑅 = ((♯‘𝑃) − 1)    &   (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖𝑗 → (𝑃𝑖) ≠ (𝑃𝑗)))       (𝜑 → ((𝑃 “ {0, 𝑅}) ∩ (𝑃 “ (1..^𝑅))) = ∅)

Theorempthd 26846* Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝜑𝑃 ∈ Word V)    &   𝑅 = ((♯‘𝑃) − 1)    &   (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖𝑗 → (𝑃𝑖) ≠ (𝑃𝑗)))    &   (♯‘𝐹) = 𝑅    &   (𝜑𝐹(Trails‘𝐺)𝑃)       (𝜑𝐹(Paths‘𝐺)𝑃)

16.3.5  Closed walks

Syntaxcclwlks 26847 Extend class notation with closed walks (of a graph).
class ClWalks

Definitiondf-clwlks 26848* Define the set of all closed walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0).

Notice that by this definition, a single vertex can be considered as a closed walk of length 0, see also 0clwlk 27253. (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)

ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

Theoremclwlks 26849* The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Revised by AV, 29-Oct-2021.)
(ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Theoremisclwlk 26850 A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremclwlkiswlk 26851 A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(ClWalks‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremclwlkwlk 26852 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))

Theoremclwlkswks 26853 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.)
(ClWalks‘𝐺) ⊆ (Walks‘𝐺)

Theoremisclwlke 26854* Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑋 → (𝐹(ClWalks‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))))

Theoremisclwlkupgr 26855* Properties of a pair of functions to be a closed walk (in a pseudograph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 11-Apr-2021.) (Revised by AV, 28-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → (𝐹(ClWalks‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))))

Theoremclwlkcomp 26856* A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝐺𝑋𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (ClWalks‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))))

Theoremclwlkcompim 26857* Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (ClWalks‘𝐺) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))

Theoremupgrclwlkcompim 26858* Implications for the properties of the components of a closed walk in a pseudograph. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 2-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (ClWalks‘𝐺)) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremclwlkcompbp 26859 Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremclwlkl1loop 26860 A closed walk of length 1 is a loop. (Contributed by AV, 22-Apr-2021.)
((Fun (iEdg‘𝐺) ∧ 𝐹(ClWalks‘𝐺)𝑃 ∧ (♯‘𝐹) = 1) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺)))

16.3.6  Circuits and cycles

Syntaxccrcts 26861 Extend class notation with circuits (in a graph).
class Circuits

Syntaxccycls 26862 Extend class notation with cycles (in a graph).
class Cycles

Definitiondf-crcts 26863* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit 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)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)

Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

Definitiondf-cycls 26864* Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)

Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

Theoremcrcts 26865* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Theoremcycls 26866* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(Cycles‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}

Theoremiscrct 26867 Sufficient and necessary conditions for a pair of functions to be a circuit (in an undirected graph): A pair of function "is" (represents) a circuit iff it is a closed trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
(𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremiscycl 26868 Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremcrctprop 26869 The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremcyclprop 26870 The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Theoremcrctisclwlk 26871 A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(ClWalks‘𝐺)𝑃)

Theoremcrctistrl 26872 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)

Theoremcrctiswlk 26873 A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremcyclispth 26874 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹(Paths‘𝐺)𝑃)

Theoremcycliswlk 26875 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremcycliscrct 26876 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹(Circuits‘𝐺)𝑃)

Theoremcyclnspth 26877 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃))

Theoremcyclispthon 26878 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘0))𝑃)

Theoremlfgrn1cycl 26879* In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≠ 1))

Theoremusgr2trlncrct 26880 In a simple graph, any trail of length 2 is not a circuit. (Contributed by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → ¬ 𝐹(Circuits‘𝐺)𝑃))

Theoremumgrn1cycl 26881 In a multigraph graph (with no loops!) there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
((𝐺 ∈ UMGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘𝐹) ≠ 1)

Theoremuspgrn2crct 26882 In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)

Theoremusgrn2cycl 26883 In a simple graph there are no cycles with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 4-Feb-2021.)
((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)

Theoremcrctcshwlkn0lem1 26884 Lemma for crctcshwlkn0 26895. (Contributed by AV, 13-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ) → ((𝐴𝐵) + 1) ≤ 𝐴)

Theoremcrctcshwlkn0lem2 26885* Lemma for crctcshwlkn0 26895. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝐽 ∈ (0...(𝑁𝑆))) → (𝑄𝐽) = (𝑃‘(𝐽 + 𝑆)))

Theoremcrctcshwlkn0lem3 26886* Lemma for crctcshwlkn0 26895. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝐽 ∈ (((𝑁𝑆) + 1)...𝑁)) → (𝑄𝐽) = (𝑃‘((𝐽 + 𝑆) − 𝑁)))

Theoremcrctcshwlkn0lem4 26887* Lemma for crctcshwlkn0 26895. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))       (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))

Theoremcrctcshwlkn0lem5 26888* Lemma for crctcshwlkn0 26895. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))       (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))

Theoremcrctcshwlkn0lem6 26889* Lemma for crctcshwlkn0 26895. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))    &   (𝜑 → (𝑃𝑁) = (𝑃‘0))       ((𝜑𝐽 = (𝑁𝑆)) → if-((𝑄𝐽) = (𝑄‘(𝐽 + 1)), (𝐼‘(𝐻𝐽)) = {(𝑄𝐽)}, {(𝑄𝐽), (𝑄‘(𝐽 + 1))} ⊆ (𝐼‘(𝐻𝐽))))

Theoremcrctcshwlkn0lem7 26890* Lemma for crctcshwlkn0 26895. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))    &   (𝜑 → (𝑃𝑁) = (𝑃‘0))       (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))

Theoremcrctcshlem1 26891 Lemma for crctcsh 26898. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)       (𝜑𝑁 ∈ ℕ0)

Theoremcrctcshlem2 26892 Lemma for crctcsh 26898. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)       (𝜑 → (♯‘𝐻) = 𝑁)

Theoremcrctcshlem3 26893* Lemma for crctcsh 26898. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))

Theoremcrctcshlem4 26894* Lemma for crctcsh 26898. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝑆 = 0) → (𝐻 = 𝐹𝑄 = 𝑃))

Theoremcrctcshwlkn0 26895* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a walk 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄)

Theoremcrctcshwlk 26896* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a walk 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(Walks‘𝐺)𝑄)

Theoremcrctcshtrl 26897* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a trail 𝐻, 𝑄. (Contributed by AV, 14-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(Trails‘𝐺)𝑄)

Theoremcrctcsh 26898* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a circuit 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.) (Proof shortened by AV, 31-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(Circuits‘𝐺)𝑄)

16.3.7  Walks as words

In general, a walk is an alternating sequence of vertices and edges, as defined in df-wlks 26676: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 13456, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in definitions df-wwlks 26904 and df-wwlksn 26905, and the representation of a walk as sequence of its vertices is called "walk as word".

Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph).

Syntaxcwwlks 26899 Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalks

Syntaxcwwlksn 26900 Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalksN

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