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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | structiedg0val 26801 | The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
⊢ 𝑆 ∈ ℕ & ⊢ (Base‘ndx) < 𝑆 & ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) | ||
Theorem | structgrssvtxlem 26802 | Lemma for structgrssvtx 26803 and structgrssiedg 26804. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 𝐸 ∈ 𝑍) & ⊢ (𝜑 → {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺) ⇒ ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺)) | ||
Theorem | structgrssvtx 26803 | The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 𝐸 ∈ 𝑍) & ⊢ (𝜑 → {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺) ⇒ ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | ||
Theorem | structgrssiedg 26804 | The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 𝐸 ∈ 𝑍) & ⊢ (𝜑 → {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⊆ 𝐺) ⇒ ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸) | ||
Theorem | struct2grstr 26805 | A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⇒ ⊢ 𝐺 Struct 〈(Base‘ndx), (.ef‘ndx)〉 | ||
Theorem | struct2grvtx 26806 | The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = 𝑉) | ||
Theorem | struct2griedg 26807 | The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = 𝐸) | ||
Theorem | graop 26808 | Any representation of a graph 𝐺 (especially as extensible structure 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉}) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.) |
⊢ 𝐻 = 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ⇒ ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻)) | ||
Theorem | grastruct 26809 | Any representation of a graph 𝐺 (especially as ordered pair 𝐺 = 〈𝑉, 𝐸〉) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.) |
⊢ 𝐻 = {〈(Base‘ndx), (Vtx‘𝐺)〉, 〈(.ef‘ndx), (iEdg‘𝐺)〉} ⇒ ⊢ ((Vtx‘𝐺) = (Vtx‘𝐻) ∧ (iEdg‘𝐺) = (iEdg‘𝐻)) | ||
Theorem | gropd 26810* | If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then the ordered pair 〈𝑉, 𝐸〉 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 11-Oct-2020.) |
⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓)) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → [〈𝑉, 𝐸〉 / 𝑔]𝜓) | ||
Theorem | grstructd 26811* | If any representation of a graph with vertices 𝑉 and edges 𝐸 has a certain property 𝜓, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.) |
⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝜓)) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) & ⊢ (𝜑 → Fun (𝑆 ∖ {∅})) & ⊢ (𝜑 → 2 ≤ (♯‘dom 𝑆)) & ⊢ (𝜑 → (Base‘𝑆) = 𝑉) & ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) ⇒ ⊢ (𝜑 → [𝑆 / 𝑔]𝜓) | ||
Theorem | gropeld 26812* | If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then the ordered pair 〈𝑉, 𝐸〉 of the set of vertices and the set of edges (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 11-Oct-2020.) |
⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔 ∈ 𝐶)) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → 〈𝑉, 𝐸〉 ∈ 𝐶) | ||
Theorem | grstructeld 26813* | If any representation of a graph with vertices 𝑉 and edges 𝐸 is an element of an arbitrary class 𝐶, then any structure with base set 𝑉 and value 𝐸 in the slot for edge functions (which is such a representation of a graph with vertices 𝑉 and edges 𝐸) is an element of this class 𝐶. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.) |
⊢ (𝜑 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = 𝐸) → 𝑔 ∈ 𝐶)) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) & ⊢ (𝜑 → Fun (𝑆 ∖ {∅})) & ⊢ (𝜑 → 2 ≤ (♯‘dom 𝑆)) & ⊢ (𝜑 → (Base‘𝑆) = 𝑉) & ⊢ (𝜑 → (.ef‘𝑆) = 𝐸) ⇒ ⊢ (𝜑 → 𝑆 ∈ 𝐶) | ||
Theorem | setsvtx 26814 | The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.) |
⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → (Vtx‘(𝐺 sSet 〈𝐼, 𝐸〉)) = (Base‘𝐺)) | ||
Theorem | setsiedg 26815 | The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → (iEdg‘(𝐺 sSet 〈𝐼, 𝐸〉)) = 𝐸) | ||
Theorem | snstrvtxval 26816 | The set of vertices of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See vtxvalsnop 26820 for the (degenerate) case where 𝑉 = (Base‘ndx). (Contributed by AV, 23-Sep-2020.) |
⊢ 𝑉 ∈ V & ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉} ⇒ ⊢ (𝑉 ≠ (Base‘ndx) → (Vtx‘𝐺) = 𝑉) | ||
Theorem | snstriedgval 26817 | The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 26821 for the (degenerate) case where 𝑉 = (Base‘ndx). (Contributed by AV, 24-Sep-2020.) |
⊢ 𝑉 ∈ V & ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉} ⇒ ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅) | ||
Theorem | vtxval0 26818 | Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.) |
⊢ (Vtx‘∅) = ∅ | ||
Theorem | iedgval0 26819 | Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.) |
⊢ (iEdg‘∅) = ∅ | ||
Theorem | vtxvalsnop 26820 | Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
⊢ 𝐵 ∈ V & ⊢ 𝐺 = {〈𝐵, 𝐵〉} ⇒ ⊢ (Vtx‘𝐺) = {𝐵} | ||
Theorem | iedgvalsnop 26821 | Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
⊢ 𝐵 ∈ V & ⊢ 𝐺 = {〈𝐵, 𝐵〉} ⇒ ⊢ (iEdg‘𝐺) = {𝐵} | ||
Theorem | vtxval3sn 26822 | Degenerated case 3 for vertices: The set of vertices of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (Vtx‘{{{𝐴}}}) = {𝐴} | ||
Theorem | iedgval3sn 26823 | Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (iEdg‘{{{𝐴}}}) = {𝐴} | ||
Theorem | vtxvalprc 26824 | Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.) |
⊢ (𝐶 ∉ V → (Vtx‘𝐶) = ∅) | ||
Theorem | iedgvalprc 26825 | Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.) |
⊢ (𝐶 ∉ V → (iEdg‘𝐶) = ∅) | ||
Syntax | cedg 26826 | Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph). |
class Edg | ||
Definition | df-edg 26827 | Define the class of edges of a graph, see also definition "E = E(G)" in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges of a class as the range of its edge function (which does not even need to be a function). Therefore, this definition could also be used for hypergraphs, pseudographs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. In some cases, this is no problem, so theorems with Edg are meaningful nevertheless (e.g., edguhgr 26908). Usually, however, this definition is used only for undirected simple (hyper-/pseudo-)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) |
⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)) | ||
Theorem | edgval 26828 | The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.) |
⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | ||
Theorem | iedgedg 26829 | An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.) |
⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺)) | ||
Theorem | edgopval 26830 | The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) |
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘〈𝑉, 𝐸〉) = ran 𝐸) | ||
Theorem | edgov 26831 | The edges of a graph represented as ordered pair, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 26830. The representation ran 𝐸 for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.) |
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉Edg𝐸) = ran 𝐸) | ||
Theorem | edgstruct 26832 | The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.) |
⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⇒ ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘𝐺) = ran 𝐸) | ||
Theorem | edgiedgb 26833* | A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) | ||
Theorem | edg0iedg0 26834 | There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) | ||
For undirected graphs, we will have the following hierarchy/taxonomy: * Undirected Hypergraph: UHGraph * Undirected loop-free graphs: ULFGraph (not defined formally yet) * Undirected simple Hypergraph: USHGraph => USHGraph ⊆ UHGraph (ushgruhgr 26848) * Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 26881) * Undirected loop-free hypergraph: ULFHGraph (not defined formally yet) => ULFHGraph ⊆ UHGraph and ULFHGraph ⊆ ULFGraph * Undirected loop-free simple hypergraph: ULFSHGraph (not defined formally yet) => ULFSHGraph ⊆ USHGraph and ULFSHGraph ⊆ ULFHGraph * Undirected simple Pseudograph: USPGraph => USPGraph ⊆ UPGraph (uspgrupgr 26955) and USPGraph ⊆ USHGraph (uspgrushgr 26954), see also uspgrupgrushgr 26956 * Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 26882) and UMGraph ⊆ ULFHGraph (umgrislfupgr 26902) * Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 26957) and USGraph ⊆ UMGraph (usgrumgr 26958) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 26963) see also usgrumgruspgr 26959 | ||
Syntax | cuhgr 26835 | Extend class notation with undirected hypergraphs. |
class UHGraph | ||
Syntax | cushgr 26836 | Extend class notation with undirected simple hypergraphs. |
class USHGraph | ||
Definition | df-uhgr 26837* | Define the class of all undirected hypergraphs. An undirected hypergraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into the power set of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 8-Oct-2020.) |
⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} | ||
Definition | df-ushgr 26838* | Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a special (non-simple, multiple, multi-) hypergraph for which the edge function 𝑒 is an injective (one-to-one) function into subsets of the set of vertices 𝑣, representing the (one or more) vertices incident to the edge. This definition corresponds to the definition of hypergraphs in section I.1 of [Bollobas] p. 7 (except that the empty set seems to be allowed to be an "edge") or section 1.10 of [Diestel] p. 27, where "E is a subset of [...] the power set of V, that is the set of all subsets of V" resp. "the elements of E are nonempty subsets (of any cardinality) of V". (Contributed by AV, 19-Jan-2020.) (Revised by AV, 8-Oct-2020.) |
⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} | ||
Theorem | isuhgr 26839 | The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) | ||
Theorem | isushgr 26840 | The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅}))) | ||
Theorem | uhgrf 26841 | The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) | ||
Theorem | ushgrf 26842 | The edge function of an undirected simple hypergraph is a one-to-one function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ USHGraph → 𝐸:dom 𝐸–1-1→(𝒫 𝑉 ∖ {∅})) | ||
Theorem | uhgrss 26843 | An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) | ||
Theorem | uhgreq12g 26844 | If two sets have the same vertices and the same edges, one set is a hypergraph iff the other set is a hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝑊 = (Vtx‘𝐻) & ⊢ 𝐹 = (iEdg‘𝐻) ⇒ ⊢ (((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) ∧ (𝑉 = 𝑊 ∧ 𝐸 = 𝐹)) → (𝐺 ∈ UHGraph ↔ 𝐻 ∈ UHGraph)) | ||
Theorem | uhgrfun 26845 | The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 15-Dec-2020.) |
⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) | ||
Theorem | uhgrn0 26846 | An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.) |
⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅) | ||
Theorem | lpvtx 26847 | The endpoints of a loop (which is an edge at index 𝐽) are two (identical) vertices 𝐴. (Contributed by AV, 1-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺)) | ||
Theorem | ushgruhgr 26848 | An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.) |
⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) | ||
Theorem | isuhgrop 26849 | The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 9-Oct-2020.) |
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (〈𝑉, 𝐸〉 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) | ||
Theorem | uhgr0e 26850 | The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → (iEdg‘𝐺) = ∅) ⇒ ⊢ (𝜑 → 𝐺 ∈ UHGraph) | ||
Theorem | uhgr0vb 26851 | The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺) = ∅)) | ||
Theorem | uhgr0 26852 | The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.) |
⊢ ∅ ∈ UHGraph | ||
Theorem | uhgrun 26853 | The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UHGraph) & ⊢ (𝜑 → 𝐻 ∈ UHGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UHGraph) | ||
Theorem | uhgrunop 26854 | The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are hypergraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UHGraph) & ⊢ (𝜑 → 𝐻 ∈ UHGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph) | ||
Theorem | ushgrun 26855 | The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ USHGraph) & ⊢ (𝜑 → 𝐻 ∈ USHGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UHGraph) | ||
Theorem | ushgrunop 26856 | The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are simple hypergraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ USHGraph) & ⊢ (𝜑 → 𝐻 ∈ USHGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph) | ||
Theorem | uhgrstrrepe 26857 | Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
⊢ 𝑉 = (Base‘𝐺) & ⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) ⇒ ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ UHGraph) | ||
Theorem | incistruhgr 26858* | An incidence structure 〈𝑃, 𝐿, 𝐼〉 "where 𝑃 is a set whose elements are called points, 𝐿 is a distinct set whose elements are called lines and 𝐼 ⊆ (𝑃 × 𝐿) is the incidence relation" (see Wikipedia "Incidence structure" (24-Oct-2020), https://en.wikipedia.org/wiki/Incidence_structure) implies an undirected hypergraph, if the incidence relation is right-total (to exclude empty edges). The points become the vertices, and the edge function is derived from the incidence relation by mapping each line ("edge") to the set of vertices incident to the line/edge. With 𝑃 = (Base‘𝑆) and by defining two new slots for lines and incidence relations (analogous to LineG and Itv) and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incidence structure is an undirected hypergraph. By choosing the incident relation appropriately, other kinds of undirected graphs (pseudographs, multigraphs, simple graphs, etc.) could be defined. (Contributed by AV, 24-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph)) | ||
Syntax | cupgr 26859 | Extend class notation with undirected pseudographs. |
class UPGraph | ||
Syntax | cumgr 26860 | Extend class notation with undirected multigraphs. |
class UMGraph | ||
Definition | df-upgr 26861* | Define the class of all undirected pseudographs. An (undirected) pseudograph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop." (in contrast to a multigraph, see df-umgr 26862). (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 24-Nov-2020.) |
⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}} | ||
Definition | df-umgr 26862* | Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set 𝑣 (of "vertices") and a function 𝑒 (representing indexed "edges") into subsets of 𝑣 of cardinality two, representing the two vertices incident to the edge. In contrast to a pseudograph, a multigraph has no loop. This is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." To provide uniform definitions for all kinds of graphs, 𝑥 ∈ (𝒫 𝑣 ∖ {∅}) is used as restriction of the class abstraction, although 𝑥 ∈ 𝒫 𝑣 would be sufficient (see prprrab 13825 and isumgrs 26875). (Contributed by AV, 24-Nov-2020.) |
⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} | ||
Theorem | isupgr 26863* | The property of being an undirected pseudograph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UPGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) | ||
Theorem | wrdupgr 26864* | The property of being an undirected pseudograph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UPGraph ↔ 𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) | ||
Theorem | upgrf 26865* | The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 26866 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | ||
Theorem | upgrfn 26866* | The edge function of an undirected pseudograph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | ||
Theorem | upgrss 26867 | An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 29-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) | ||
Theorem | upgrn0 26868 | An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ≠ ∅) | ||
Theorem | upgrle 26869 | An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) | ||
Theorem | upgrfi 26870 | An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ Fin) | ||
Theorem | upgrex 26871* | An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦}) | ||
Theorem | upgrbi 26872* | Show that an unordered pair is a valid edge in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 28-Feb-2021.) |
⊢ 𝑋 ∈ 𝑉 & ⊢ 𝑌 ∈ 𝑉 ⇒ ⊢ {𝑋, 𝑌} ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} | ||
Theorem | upgrop 26873 | A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.) |
⊢ (𝐺 ∈ UPGraph → 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ UPGraph) | ||
Theorem | isumgr 26874* | The property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) = 2})) | ||
Theorem | isumgrs 26875* | The simplified property of being an undirected multigraph. (Contributed by AV, 24-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (𝐺 ∈ UMGraph ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) | ||
Theorem | wrdumgr 26876* | The property of being an undirected multigraph, expressing the edges as "words". (Contributed by AV, 24-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋) → (𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})) | ||
Theorem | umgrf 26877* | The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn 26878 without explicitly specified domain of the edge function. (Contributed by AV, 24-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UMGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | ||
Theorem | umgrfn 26878* | The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by AV, 24-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) | ||
Theorem | umgredg2 26879 | An edge of a multigraph has exactly two ends. (Contributed by AV, 24-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → (♯‘(𝐸‘𝑋)) = 2) | ||
Theorem | umgrbi 26880* | Show that an unordered pair is a valid edge in a multigraph. (Contributed by AV, 9-Mar-2021.) |
⊢ 𝑋 ∈ 𝑉 & ⊢ 𝑌 ∈ 𝑉 & ⊢ 𝑋 ≠ 𝑌 ⇒ ⊢ {𝑋, 𝑌} ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} | ||
Theorem | upgruhgr 26881 | An undirected pseudograph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.) |
⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | ||
Theorem | umgrupgr 26882 | An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.) |
⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | ||
Theorem | umgruhgr 26883 | An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.) |
⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | ||
Theorem | upgrle2 26884 | An edge of an undirected pseudograph has at most two ends. (Contributed by AV, 6-Feb-2021.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2) | ||
Theorem | umgrnloopv 26885 | In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 11-Dec-2020.) |
⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑀 ∈ 𝑊) → ((𝐸‘𝑋) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁)) | ||
Theorem | umgredgprv 26886 | In a multigraph, an edge is an unordered pair of vertices. This theorem would not hold for arbitrary hyper-/pseudographs since either 𝑀 or 𝑁 could be proper classes ((𝐸‘𝑋) would be a loop in this case), which are no vertices of course. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.) |
⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐸) → ((𝐸‘𝑋) = {𝑀, 𝑁} → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))) | ||
Theorem | umgrnloop 26887* | In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 11-Dec-2020.) |
⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UMGraph → (∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = {𝑀, 𝑁} → 𝑀 ≠ 𝑁)) | ||
Theorem | umgrnloop0 26888* | A multigraph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 11-Dec-2020.) |
⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UMGraph → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) = {𝑈}} = ∅) | ||
Theorem | umgr0e 26889 | The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → (iEdg‘𝐺) = ∅) ⇒ ⊢ (𝜑 → 𝐺 ∈ UMGraph) | ||
Theorem | upgr0e 26890 | The empty graph, with vertices but no edges, is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → (iEdg‘𝐺) = ∅) ⇒ ⊢ (𝜑 → 𝐺 ∈ UPGraph) | ||
Theorem | upgr1elem 26891* | Lemma for upgr1e 26892 and uspgr1e 27020. (Contributed by AV, 16-Oct-2020.) |
⊢ (𝜑 → {𝐵, 𝐶} ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → {{𝐵, 𝐶}} ⊆ {𝑥 ∈ (𝑆 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | ||
Theorem | upgr1e 26892 | A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1e 27020. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) ⇒ ⊢ (𝜑 → 𝐺 ∈ UPGraph) | ||
Theorem | upgr0eop 26893 | The empty graph, with vertices but no edges, is a pseudograph. The empty graph is actually a simple graph, see usgr0eop 27022, and therefore also a multigraph (𝐺 ∈ UMGraph). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.) |
⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) | ||
Theorem | upgr1eop 26894 | A pseudograph with one edge. Such a graph is actually a simple pseudograph, see uspgr1eop 27023. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ UPGraph) | ||
Theorem | upgr0eopALT 26895 | Alternate proof of upgr0eop 26893, using the general theorem gropeld 26812 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr0eop 26893). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ UPGraph) | ||
Theorem | upgr1eopALT 26896 | Alternate proof of upgr1eop 26894, using the general theorem gropeld 26812 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr1eop 26894). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ UPGraph) | ||
Theorem | upgrun 26897 | The union 𝑈 of two pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UPGraph) & ⊢ (𝜑 → 𝐻 ∈ UPGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UPGraph) | ||
Theorem | upgrunop 26898 | The union of two pseudographs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are pseudographs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
⊢ (𝜑 → 𝐺 ∈ UPGraph) & ⊢ (𝜑 → 𝐻 ∈ UPGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) | ||
Theorem | umgrun 26899 | The union 𝑈 of two multigraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a multigraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐺 ∈ UMGraph) & ⊢ (𝜑 → 𝐻 ∈ UMGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UMGraph) | ||
Theorem | umgrunop 26900 | The union of two multigraphs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are multigraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a multigraph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐺 ∈ UMGraph) & ⊢ (𝜑 → 𝐻 ∈ UMGraph) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UMGraph) |
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