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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | opiedgov 28701 | The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.) |
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉iEdg𝐸) = 𝐸) | ||
Theorem | opvtxfvi 28702 | The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.) |
⊢ 𝑉 ∈ V & ⊢ 𝐸 ∈ V ⇒ ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 | ||
Theorem | opiedgfvi 28703 | The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.) |
⊢ 𝑉 ∈ V & ⊢ 𝐸 ∈ V ⇒ ⊢ (iEdg‘〈𝑉, 𝐸〉) = 𝐸 | ||
Theorem | funvtxdmge2val 28704 | The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
Theorem | funiedgdmge2val 28705 | The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | ||
Theorem | funvtxdm2val 28706 | The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
Theorem | funiedgdm2val 28707 | The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺)) | ||
Theorem | funvtxval0 28708 | The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ 𝑆 ∈ V ⇒ ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
Theorem | basvtxval 28709 | The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.) |
⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺)) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 〈(Base‘ndx), 𝑉〉 ∈ 𝐺) ⇒ ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | ||
Theorem | edgfiedgval 28710 | The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.) |
⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐺)) & ⊢ (𝜑 → 𝐸 ∈ 𝑌) & ⊢ (𝜑 → 〈(.ef‘ndx), 𝐸〉 ∈ 𝐺) ⇒ ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸) | ||
Theorem | funvtxval 28711 | The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ ((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
Theorem | funiedgval 28712 | The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
⊢ ((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺)) | ||
Theorem | structvtxvallem 28713 | Lemma for structvtxval 28714 and structiedg0val 28715. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.) |
⊢ 𝑆 ∈ ℕ & ⊢ (Base‘ndx) < 𝑆 & ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺)) | ||
Theorem | structvtxval 28714 | The set of vertices of an extensible structure with a base set and another slot. (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) → (Vtx‘𝐺) = 𝑉) | ||
Theorem | structiedg0val 28715 | 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 28716 | Lemma for structgrssvtx 28717 and structgrssiedg 28718. (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 28717 | 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 28718 | 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 28719 | 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 28720 | 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 28721 | 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 28722 | 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 28723 | 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 28724* | 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 28725* | 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 28726* | 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 28727* | 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 28728 | 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 28729 | 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 28730 | 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 28734 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 28731 | 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 28735 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 28732 | 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 28733 | 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 28734 | 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 28735 | 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 28736 | 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 28737 | 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 28738 | 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 28739 | 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 28740 | Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph). |
class Edg | ||
Definition | df-edg 28741 | 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 28822). 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 28742 | 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 28743 | 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 28744 | 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 28745 | 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 28744. 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 28746 | 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 28747* | 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 28748 | 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 28762) * Undirected Pseudograph: UPGraph => UPGraph ⊆ UHGraph (upgruhgr 28795) * 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 28869) and USPGraph ⊆ USHGraph (uspgrushgr 28868), see also uspgrupgrushgr 28870 * Undirected Muligraph: UMGraph => UMGraph ⊆ UPGraph (umgrupgr 28796) and UMGraph ⊆ ULFHGraph (umgrislfupgr 28816) * Undirected simple Graph: USGraph => USGraph ⊆ USPGraph (usgruspgr 28871) and USGraph ⊆ UMGraph (usgrumgr 28872) and USGraph ⊆ ULFSHGraph (usgrislfuspgr 28877) see also usgrumgruspgr 28873 | ||
Syntax | cuhgr 28749 | Extend class notation with undirected hypergraphs. |
class UHGraph | ||
Syntax | cushgr 28750 | Extend class notation with undirected simple hypergraphs. |
class USHGraph | ||
Definition | df-uhgr 28751* | 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 28752* | 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 28753 | 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 28754 | 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 28755 | 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 28756 | 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 28757 | 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 28758 | 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 28759 | 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 28760 | 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 28761 | 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 28762 | 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 28763 | 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 28764 | 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 28765 | 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 28766 | The null graph represented by an empty set is a hypergraph. (Contributed by AV, 9-Oct-2020.) |
⊢ ∅ ∈ UHGraph | ||
Theorem | uhgrun 28767 | 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 28768 | 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 28769 | 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 28770 | 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 28771 | 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 28772* | 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 28773 | Extend class notation with undirected pseudographs. |
class UPGraph | ||
Syntax | cumgr 28774 | Extend class notation with undirected multigraphs. |
class UMGraph | ||
Definition | df-upgr 28775* | 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 28776). (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 28776* | 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 14441 and isumgrs 28789). (Contributed by AV, 24-Nov-2020.) |
⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (♯‘𝑥) = 2}} | ||
Theorem | isupgr 28777* | 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 28778* | 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 28779* | The edge function of an undirected pseudograph is a function into unordered pairs of vertices. Version of upgrfn 28780 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 28780* | 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 28781 | 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 28782 | 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 28783 | 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 28784 | 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 28785* | 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 28786* | 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 28787 | A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.) |
⊢ (𝐺 ∈ UPGraph → 〈(Vtx‘𝐺), (iEdg‘𝐺)〉 ∈ UPGraph) | ||
Theorem | isumgr 28788* | 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 28789* | 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 28790* | 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 28791* | The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgrfn 28792 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 28792* | 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 28793 | 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 28794* | 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 28795 | 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 28796 | An undirected multigraph is an undirected pseudograph. (Contributed by AV, 25-Nov-2020.) |
⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | ||
Theorem | umgruhgr 28797 | An undirected multigraph is an undirected hypergraph. (Contributed by AV, 26-Nov-2020.) |
⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph) | ||
Theorem | upgrle2 28798 | 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 28799 | 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 28800 | 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))) |
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