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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisspthonpth 26701 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 26702 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 26703 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 26704 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 26705 Lemma 1 for uhgrwkspth 26707. (Contributed by AV, 25-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 1) → Fun 𝐹)
 
Theoremuhgrwkspthlem2 26706 Lemma 2 for uhgrwkspth 26707. (Contributed by AV, 25-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ ((#‘𝐹) = 1 ∧ 𝐴𝐵) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → Fun 𝑃)
 
Theoremuhgrwkspth 26707 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 26708 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 26709 Lemma 1 for usgr2wlkspth 26711. (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 26710 Lemma 2 for usgr2wlkspth 26711. (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 26711 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 26712 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 26713 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 26714 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 26715* Lemma for usgr2pth 26716. (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 26716* 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 26717* 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 26718* Lemma 1 for pthd 26721. (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 26719* Lemma for pthdlem2 26720. (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 26720* Lemma 2 for pthd 26721. (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 26721* 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 26722 Extend class notation with closed walks (of a graph).
class ClWalks
 
Definitiondf-clwlks 26723* 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 27108. (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 26724* 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 26725 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 26726 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 26727 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 26728 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 26729* 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 26730* 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 26731* 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 26732* 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 26733* 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))))
 
Theoremclwlkl1loop 26734 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 26735 Extend class notation with circuits (in a graph).
class Circuits
 
Syntaxccycls 26736 Extend class notation with cycles (in a graph).
class Cycles
 
Definitiondf-crcts 26737* 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 26738* 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 26739* 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 26740* 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 26741 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 26742 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 26743 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 26744 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 26745 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 26746 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 26747 A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
 
Theoremcyclispth 26748 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 26749 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 26750 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 26751 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 26752 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 26753* 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 26754 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 26755 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 26756 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 26757 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 26758 Lemma for crctcshwlkn0 26769. (Contributed by AV, 13-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ) → ((𝐴𝐵) + 1) ≤ 𝐴)
 
Theoremcrctcshwlkn0lem2 26759* Lemma for crctcshwlkn0 26769. (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 26760* Lemma for crctcshwlkn0 26769. (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 26761* Lemma for crctcshwlkn0 26769. (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 26762* Lemma for crctcshwlkn0 26769. (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 26763* Lemma for crctcshwlkn0 26769. (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 26764* Lemma for crctcshwlkn0 26769. (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 26765 Lemma for crctcsh 26772. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)       (𝜑𝑁 ∈ ℕ0)
 
Theoremcrctcshlem2 26766 Lemma for crctcsh 26772. (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 26767* Lemma for crctcsh 26772. (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 26768* Lemma for crctcsh 26772. (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 26769* 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 26770* 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 26771* 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 26772* 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 26551: 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 13331, 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 26778 and df-wwlksn 26779, 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 26773 Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalks
 
Syntaxcwwlksn 26774 Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalksN
 
Syntaxcwwlksnon 26775 Extend class notation with walks between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WWalksNOn
 
Syntaxcwwspthsn 26776 Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN
 
Syntaxcwwspthsnon 26777 Extend class notation with simple paths between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WSPathsNOn
 
Definitiondf-wwlks 26778* Define the set of all walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 26551. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 26572. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
 
Definitiondf-wwlksn 26779* Define the set of all walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 26551. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
 
Definitiondf-wwlksnon 26780* Define the collection of walks of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
 
Definitiondf-wspthsn 26781* Define the collection of simple paths of a fixed length as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
 
Definitiondf-wspthsnon 26782* Define the collection of simple paths of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
 
Theoremwwlks 26783* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (WWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}
 
Theoremiswwlks 26784* A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlksn 26785* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
 
Theoremiswwlksn 26786 A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))))
 
Theoremwwlksnprcl 26787 Derivation of the length of a word 𝑊 whose concatenation with a singleton word represents a walk of a fixed length 𝑁 (a partial reverse closure theorem). (Contributed by AV, 4-Mar-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (#‘𝑊) = 𝑁))
 
Theoremiswwlksnx 26788* Properties of a word to represent a walk of a fixed length, definition of WWalks expanded. (Contributed by AV, 28-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (#‘𝑊) = (𝑁 + 1))))
 
Theoremwwlkbp 26789 Basic properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (WWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉))
 
Theoremwwlknbp 26790 Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 20-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
 
Theoremwwlknp 26791* Properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlknbp1 26792 Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
 
Theoremwwlknvtx 26793* The symbols of a word 𝑊 representing a walk of a fixed length 𝑁 are vertices. (Contributed by AV, 16-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑖 ∈ (0...𝑁)(𝑊𝑖) ∈ (Vtx‘𝐺))
 
Theoremwwlknbp2OLD 26794 Obsolete version of wwlknbp1 26792 as of 14-Mar-2022. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 12-Apr-2021.) (Proof shortened by AV, 8-Mar-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
 
Theoremwwlknllvtx 26795 If a word 𝑊 represents a walk of a fixed length 𝑁, then the first and the last symbol of the word is a vertex. (Contributed by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ 𝑉 ∧ (𝑊𝑁) ∈ 𝑉))
 
Theoremwwlknlsw 26796 If a word represents a walk of a fixed length, then the last symbol of the word is the endvertex of the walk. (Contributed by AV, 8-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊𝑁) = ( lastS ‘𝑊))
 
Theoremwspthsn 26797* The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
 
Theoremiswspthn 26798* An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 
Theoremwspthnp 26799* Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 
Theoremwwlksnon 26800* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
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