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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ewlkprop 26901* | Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | ewlkinedg 26902 | The intersection (common vertices) of two adjacent edges in an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝐾 ∈ (1..^(♯‘𝐹))) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))) | ||
Theorem | ewlkle 26903 | An s-walk of edges is also a t-walk of edges if t <_ s. (Contributed by AV, 4-Jan-2021.) |
⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ0* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇)) | ||
Theorem | upgrewlkle2 26904 | In a pseudograph, there is no s-walk of edges of length greater than 1 with s>2. (Contributed by AV, 4-Jan-2021.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2) | ||
Theorem | wkslem1 26905 | Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵))))) | ||
Theorem | wkslem2 26906 | Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵))))) | ||
Theorem | wksfval 26907* | The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (Walks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))}) | ||
Theorem | iswlk 26908* | Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | wlkprop 26909* | Properties of a walk. (Contributed by AV, 5-Nov-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))) | ||
Theorem | wlkv 26910 | The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
Theorem | iswlkg 26911* | Generalization of iswlk 26908: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | wlkf 26912 | The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) | ||
Theorem | wlkcl 26913 | A walk has length ♯(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 27575. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0) | ||
Theorem | wlkp 26914 | The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉) | ||
Theorem | wlkpwrd 26915 | The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word 𝑉) | ||
Theorem | wlklenvp1 26916 | The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1)) | ||
Theorem | wksv 26917* | The class of walks is a set. (Contributed by AV, 15-Jan-2021.) |
⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | ||
Theorem | wlkn0 26918 | The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅) | ||
Theorem | wlklenvm1 26919 | The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1)) | ||
Theorem | ifpsnprss 26920 | Lemma for wlkvtxeledg 26921: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.) |
⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) | ||
Theorem | wlkvtxeledg 26921* | Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) | ||
Theorem | wlkvtxiedg 26922* | The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ ran 𝐼{(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒) | ||
Theorem | relwlk 26923 | The set (Walks‘𝐺) of all walks on 𝐺 is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.) |
⊢ Rel (Walks‘𝐺) | ||
Theorem | wlkvv 26924 | If there is at least one walk in the graph, all walks are in the universal class of ordered pairs. (Contributed by AV, 2-Jan-2021.) |
⊢ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) → 𝑊 ∈ (V × V)) | ||
Theorem | wlkop 26925 | A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | ||
Theorem | wlkcpr 26926 | A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊)) | ||
Theorem | wlk2f 26927* | If there is a walk 𝑊 there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
⊢ (𝑊 ∈ (Walks‘𝐺) → ∃𝑓∃𝑝 𝑓(Walks‘𝐺)𝑝) | ||
Theorem | wlkcomp 26928* | A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘𝐺) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | wlkcompim 26929* | Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))) | ||
Theorem | wlkelwrd 26930 | The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉)) | ||
Theorem | wlkeq 26931* | Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.) |
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥)))) | ||
Theorem | edginwlk 26932 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((Fun 𝐼 ∧ 𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸) | ||
Theorem | upgredginwlk 26933 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝐾 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸)) | ||
Theorem | iedginwlk 26934 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((Fun 𝐼 ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑋)) ∈ ran 𝐼) | ||
Theorem | wlkl1loop 26935 | A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.) |
⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺)) | ||
Theorem | wlk1walk 26936* | A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))) | ||
Theorem | wlk1ewlk 26937 | A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ (𝐺 EdgWalks 1)) | ||
Theorem | upgriswlk 26938* | Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgrwlkedg 26939* | The edges of a walk in a pseudograph join exactly the two corresponding adjacent vertices in the walk. (Contributed by AV, 27-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) | ||
Theorem | upgrwlkcompim 26940* | Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 14-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = (1st ‘𝑊) & ⊢ 𝑃 = (2nd ‘𝑊) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) | ||
Theorem | wlkvtxedg 26941* | The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ 𝐸 {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒) | ||
Theorem | upgrwlkvtxedg 26942* | The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸) | ||
Theorem | uspgr2wlkeq 26943* | Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.) |
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))) | ||
Theorem | uspgr2wlkeq2 26944 | Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.) |
⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘𝐵)) = 𝑁)) → ((2nd ‘𝐴) = (2nd ‘𝐵) → 𝐴 = 𝐵)) | ||
Theorem | uspgr2wlkeqi 26945 | Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 6-May-2021.) |
⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) → 𝐴 = 𝐵) | ||
Theorem | umgrwlknloop 26946* | In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.) |
⊢ ((𝐺 ∈ UMGraph ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Theorem | wlkRes 26947* | Restrictions of walks (i.e. special kinds of walks, for example paths - see pthsfval 27023) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 30-Dec-2020.) (Proof shortened by AV, 15-Jan-2021.) |
⊢ (𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) ⇒ ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V | ||
Theorem | wlkv0 26948 | If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ (((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (Walks‘𝐺)) → ((1st ‘𝑊) = ∅ ∧ (2nd ‘𝑊) = ∅)) | ||
Theorem | g0wlk0 26949 | There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ ((Vtx‘𝐺) = ∅ → (Walks‘𝐺) = ∅) | ||
Theorem | 0wlk0 26950 | There is no walk for the empty set, i.e. in a null graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ (Walks‘∅) = ∅ | ||
Theorem | wlk0prc 26951 | There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.) |
⊢ ((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (Walks‘𝐺) = ∅) | ||
Theorem | wlklenvclwlk 26952 | The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) |
⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊)) → (〈𝐹, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊))) | ||
Theorem | wlkson 26953* | The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)}) | ||
Theorem | iswlkon 26954 | Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 31-Dec-2020.) (Revised by AV, 22-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) | ||
Theorem | wlkonprop 26955 | Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 16-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵))) | ||
Theorem | wlkpvtx 26956 | A walk connects vertices. (Contributed by AV, 22-Feb-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝑁 ∈ (0...(♯‘𝐹)) → (𝑃‘𝑁) ∈ 𝑉)) | ||
Theorem | wlkepvtx 26957 | The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉)) | ||
Theorem | wlkoniswlk 26958 | A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 2-Jan-2021.) |
⊢ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | wlkonwlk 26959 | A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.) |
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃) | ||
Theorem | wlkonwlk1l 26960 | A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 22-Mar-2021.) |
⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) ⇒ ⊢ (𝜑 → 𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃) | ||
Theorem | wlksoneq1eq2 26961 | Two walks with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.) |
⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐻(𝐶(WalksOn‘𝐺)𝐷)𝑃) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | wlkonl1iedg 26962* | If there is a walk between two vertices 𝐴 and 𝐵 at least of length 1, then the start vertex 𝐴 is incident with an edge. (Contributed by AV, 4-Apr-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ (♯‘𝐹) ≠ 0) → ∃𝑒 ∈ ran 𝐼 𝐴 ∈ 𝑒) | ||
Theorem | wlkon2n0 26963 | The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.) |
⊢ ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ 𝐴 ≠ 𝐵) → (♯‘𝐹) ≠ 0) | ||
Theorem | 2wlklem 26964* | Lemma for theorems for walks of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) |
⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) | ||
Theorem | upgr2wlk 26965 | Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) | ||
Theorem | wlkreslem0OLD 26966 | Obsolete as of 30-Nov-2022. (Contributed by AV, 5-Mar-2021.) (Proof shortened by AV, 3-Nov-2022.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁) | ||
Theorem | wlkreslem0OLDOLD 26967 | Obsolete proof of wlkreslem0OLD 26966 as of 12-Oct-2022. (Contributed by AV, 5-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐹 ∈ Word 𝑆 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁) | ||
Theorem | wlkreslem 26968 | Lemma for wlkres 26969. (Contributed by AV, 5-Mar-2021.) (Revised by AV, 30-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) ⇒ ⊢ (𝜑 → 𝑆 ∈ V) | ||
Theorem | wlkres 26969 | The restriction 〈𝐻, 𝑄〉 of a walk 〈𝐹, 𝑃〉 to an initial segment of the walk (of length 𝑁) forms a walk on the subgraph 𝑆 consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 27581. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 prefix 𝑁) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) ⇒ ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) | ||
Theorem | wlkreslemOLD 26970 | Obsolete version of wlkreslem 26968 as of 30-Nov-2022. (Contributed by AV, 5-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) ⇒ ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) | ||
Theorem | wlkresOLD 26971 | Obsolete version of wlkres 26969 as of 30-Nov-2022. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) ⇒ ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) | ||
Theorem | redwlklem 26972 | Lemma for redwlk 26973. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.) |
⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (♯‘𝐹) ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → (𝑃 ↾ (0..^(♯‘𝐹))):(0...(♯‘(𝐹 ↾ (0..^((♯‘𝐹) − 1)))))⟶𝑉) | ||
Theorem | redwlk 26973 | A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.) |
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 1 ≤ (♯‘𝐹)) → (𝐹 ↾ (0..^((♯‘𝐹) − 1)))(Walks‘𝐺)(𝑃 ↾ (0..^(♯‘𝐹)))) | ||
Theorem | wlkp1lem1 26974 | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) ⇒ ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃) | ||
Theorem | wlkp1lem2 26975 | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) ⇒ ⊢ (𝜑 → (♯‘𝐻) = (𝑁 + 1)) | ||
Theorem | wlkp1lem3 26976 | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) ⇒ ⊢ (𝜑 → ((iEdg‘𝑆)‘(𝐻‘𝑁)) = ((𝐼 ∪ {〈𝐵, 𝐸〉})‘𝐵)) | ||
Theorem | wlkp1lem4 26977 | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) ⇒ ⊢ (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V)) | ||
Theorem | wlkp1lem5 26978* | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄‘𝑘) = (𝑃‘𝑘)) | ||
Theorem | wlkp1lem6 26979* | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄‘𝑘) = (𝑃‘𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻‘𝑘)) = (𝐼‘(𝐹‘𝑘)))) | ||
Theorem | wlkp1lem7 26980 | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) ⇒ ⊢ (𝜑 → {(𝑄‘𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑁))) | ||
Theorem | wlkp1lem8 26981* | Lemma for wlkp1 26982. (Contributed by AV, 6-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐻))if-((𝑄‘𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻‘𝑘)) = {(𝑄‘𝑘)}, {(𝑄‘𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻‘𝑘)))) | ||
Theorem | wlkp1 26982 | Append one path segment (edge) 𝐸 from vertex (𝑃‘𝑁) to a vertex 𝐶 to a walk 〈𝐹, 𝑃〉 to become a walk 〈𝐻, 𝑄〉 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 27582. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 18-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) & ⊢ 𝑁 = (♯‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {〈𝐵, 𝐸〉})) & ⊢ 𝐻 = (𝐹 ∪ {〈𝑁, 𝐵〉}) & ⊢ 𝑄 = (𝑃 ∪ {〈(𝑁 + 1), 𝐶〉}) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) ⇒ ⊢ (𝜑 → 𝐻(Walks‘𝑆)𝑄) | ||
Theorem | wlkdlem1 26983* | Lemma 1 for wlkd 26987. (Contributed by AV, 7-Feb-2021.) |
⊢ (𝜑 → 𝑃 ∈ Word V) & ⊢ (𝜑 → 𝐹 ∈ Word V) & ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) & ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶𝑉) | ||
Theorem | wlkdlem2 26984* | Lemma 2 for wlkd 26987. (Contributed by AV, 7-Feb-2021.) |
⊢ (𝜑 → 𝑃 ∈ Word V) & ⊢ (𝜑 → 𝐹 ∈ Word V) & ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → (((♯‘𝐹) ∈ ℕ → (𝑃‘(♯‘𝐹)) ∈ (𝐼‘(𝐹‘((♯‘𝐹) − 1)))) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ∈ (𝐼‘(𝐹‘𝑘)))) | ||
Theorem | wlkdlem3 26985* | Lemma 3 for wlkd 26987. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.) |
⊢ (𝜑 → 𝑃 ∈ Word V) & ⊢ (𝜑 → 𝐹 ∈ Word V) & ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) | ||
Theorem | wlkdlem4 26986* | Lemma 4 for wlkd 26987. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 23-Jan-2021.) |
⊢ (𝜑 → 𝑃 ∈ Word V) & ⊢ (𝜑 → 𝐹 ∈ Word V) & ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) | ||
Theorem | wlkd 26987* | Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.) |
⊢ (𝜑 → 𝑃 ∈ Word V) & ⊢ (𝜑 → 𝐹 ∈ Word V) & ⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1)) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | lfgrwlkprop 26988* | Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Theorem | lfgriswlk 26989* | Conditions for a pair of functions to be a walk in a loop-free graph. (Contributed by AV, 28-Jan-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))) | ||
Theorem | lfgrwlknloop 26990* | In a loop-free graph, each walk has no loops! (Contributed by AV, 2-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Syntax | ctrls 26991 | Extend class notation with trails (within a graph). |
class Trails | ||
Syntax | ctrlson 26992 | Extend class notation with trails between two vertices (within a graph). |
class TrailsOn | ||
Definition | df-trls 26993* |
Define the set of all Trails (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct. According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5. Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail 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). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ Trails = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) | ||
Definition | df-trlson 26994* | Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(Trails‘𝑔)𝑝)})) | ||
Theorem | reltrls 26995 | The set (Trails‘𝐺) of all trails on 𝐺 is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.) |
⊢ Rel (Trails‘𝐺) | ||
Theorem | trlsfval 26996* | The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.) |
⊢ (Trails‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)} | ||
Theorem | istrl 26997 | Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.) |
⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | ||
Theorem | trliswlk 26998 | A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.) |
⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | trlf1 26999 | The enumeration 𝐹 of a trail 〈𝐹, 𝑃〉 is injective. (Contributed by AV, 20-Feb-2021.) (Proof shortened by AV, 29-Oct-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) | ||
Theorem | trlreslem 27000 | Lemma for trlres 27001. Formerly part of proof of eupthres 27581. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) & ⊢ 𝐻 = (𝐹 prefix 𝑁) ⇒ ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
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