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Type | Label | Description |
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Statement | ||
Theorem | subgrv 29301 | If a class is a subgraph of another class, both classes are sets. (Contributed by AV, 16-Nov-2020.) |
⊢ (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V)) | ||
Theorem | issubgr 29302 | The property of a set to be a subgraph of another set. (Contributed by AV, 16-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐴 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝑆) & ⊢ 𝐵 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝑆) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) | ||
Theorem | issubgr2 29303 | The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐴 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝑆) & ⊢ 𝐵 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝑆) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))) | ||
Theorem | subgrprop 29304 | The properties of a subgraph. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐴 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝑆) & ⊢ 𝐵 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝑆) ⇒ ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)) | ||
Theorem | subgrprop2 29305 | The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐴 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝑆) & ⊢ 𝐵 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝑆) ⇒ ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉)) | ||
Theorem | uhgrissubgr 29306 | The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐴 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝑆) & ⊢ 𝐵 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ UHGraph) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵))) | ||
Theorem | subgrprop3 29307 | The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐴 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝑆) & ⊢ 𝐵 = (Edg‘𝐺) ⇒ ⊢ (𝑆 SubGraph 𝐺 → (𝑉 ⊆ 𝐴 ∧ 𝐸 ⊆ 𝐵)) | ||
Theorem | egrsubgr 29308 | An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.) |
⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺) | ||
Theorem | 0grsubgr 29309 | The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) |
⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺) | ||
Theorem | 0uhgrsubgr 29310 | The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺) | ||
Theorem | uhgrsubgrself 29311 | A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
⊢ (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺) | ||
Theorem | subgrfun 29312 | The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.) |
⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) | ||
Theorem | subgruhgrfun 29313 | The edge function of a subgraph of a hypergraph is a function. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 20-Nov-2020.) |
⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) | ||
Theorem | subgreldmiedg 29314 | An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.) |
⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺)) | ||
Theorem | subgruhgredgd 29315 | An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐼 = (iEdg‘𝑆) & ⊢ (𝜑 → 𝐺 ∈ UHGraph) & ⊢ (𝜑 → 𝑆 SubGraph 𝐺) & ⊢ (𝜑 → 𝑋 ∈ dom 𝐼) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) ∈ (𝒫 𝑉 ∖ {∅})) | ||
Theorem | subumgredg2 29316* | An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝑆) & ⊢ 𝐼 = (iEdg‘𝑆) ⇒ ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}) | ||
Theorem | subuhgr 29317 | A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph) | ||
Theorem | subupgr 29318 | A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph) | ||
Theorem | subumgr 29319 | A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.) |
⊢ ((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph) | ||
Theorem | subusgr 29320 | A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.) |
⊢ ((𝐺 ∈ USGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ USGraph) | ||
Theorem | uhgrspansubgrlem 29321 | Lemma for uhgrspansubgr 29322: The edges of the graph 𝑆 obtained by removing some edges of a hypergraph 𝐺 are subsets of its vertices (a spanning subgraph, see comment for uhgrspansubgr 29322. (Contributed by AV, 18-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ UHGraph) ⇒ ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) | ||
Theorem | uhgrspansubgr 29322 | A spanning subgraph 𝑆 of a hypergraph 𝐺 is actually a subgraph of 𝐺. A subgraph 𝑆 of a graph 𝐺 which has the same vertices as 𝐺 and is obtained by removing some edges of 𝐺 is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ UHGraph) ⇒ ⊢ (𝜑 → 𝑆 SubGraph 𝐺) | ||
Theorem | uhgrspan 29323 | A spanning subgraph 𝑆 of a hypergraph 𝐺 is a hypergraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ UHGraph) ⇒ ⊢ (𝜑 → 𝑆 ∈ UHGraph) | ||
Theorem | upgrspan 29324 | A spanning subgraph 𝑆 of a pseudograph 𝐺 is a pseudograph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ UPGraph) ⇒ ⊢ (𝜑 → 𝑆 ∈ UPGraph) | ||
Theorem | umgrspan 29325 | A spanning subgraph 𝑆 of a multigraph 𝐺 is a multigraph. (Contributed by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ UMGraph) ⇒ ⊢ (𝜑 → 𝑆 ∈ UMGraph) | ||
Theorem | usgrspan 29326 | A spanning subgraph 𝑆 of a simple graph 𝐺 is a simple graph. (Contributed by AV, 15-Oct-2020.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 18-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ USGraph) ⇒ ⊢ (𝜑 → 𝑆 ∈ USGraph) | ||
Theorem | uhgrspanop 29327 | A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UHGraph) | ||
Theorem | upgrspanop 29328 | A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UPGraph) | ||
Theorem | umgrspanop 29329 | A spanning subgraph of a multigraph represented by an ordered pair is a multigraph. (Contributed by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UMGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UMGraph) | ||
Theorem | usgrspanop 29330 | A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ USGraph) | ||
Theorem | uhgrspan1lem1 29331 | Lemma 1 for uhgrspan1 29334. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)} ⇒ ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) | ||
Theorem | uhgrspan1lem2 29332 | Lemma 2 for uhgrspan1 29334. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉 ⇒ ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) | ||
Theorem | uhgrspan1lem3 29333 | Lemma 3 for uhgrspan1 29334. (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉 ⇒ ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹) | ||
Theorem | uhgrspan1 29334* | The induced subgraph 𝑆 of a hypergraph 𝐺 obtained by removing one vertex is actually a subgraph of 𝐺. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graph that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 SubGraph 𝐺) | ||
Theorem | upgrreslem 29335* | Lemma for upgrres 29337. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (♯‘𝑝) ≤ 2}) | ||
Theorem | umgrreslem 29336* | Lemma for umgrres 29338 and usgrres 29339. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) | ||
Theorem | upgrres 29337* | A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 29334) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph) | ||
Theorem | umgrres 29338* | A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 29334) is a multigraph. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) | ||
Theorem | usgrres 29339* | A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 29334) is a simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 19-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) | ||
Theorem | upgrres1lem1 29340* | Lemma 1 for upgrres1 29344. (Contributed by AV, 7-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⇒ ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) | ||
Theorem | umgrres1lem 29341* | Lemma for umgrres1 29345. (Contributed by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran ( I ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (♯‘𝑝) = 2}) | ||
Theorem | upgrres1lem2 29342* | Lemma 2 for upgrres1 29344. (Contributed by AV, 7-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) | ||
Theorem | upgrres1lem3 29343* | Lemma 3 for upgrres1 29344. (Contributed by AV, 7-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) | ||
Theorem | upgrres1 29344* | A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 29299 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph) | ||
Theorem | umgrres1 29345* | A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 29299 since the domains of the edge functions may not be compatible. (Contributed by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph) | ||
Theorem | usgrres1 29346* | Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 29299 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph) | ||
Syntax | cfusgr 29347 | Extend class notation with finite simple graphs. |
class FinUSGraph | ||
Definition | df-fusgr 29348 | Define the class of all finite undirected simple graphs without loops (called "finite simple graphs" in the following). A finite simple graph is an undirected simple graph of finite order, i.e. with a finite set of vertices. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} | ||
Theorem | isfusgr 29349 | The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) | ||
Theorem | fusgrvtxfi 29350 | A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph → 𝑉 ∈ Fin) | ||
Theorem | isfusgrf1 29351* | The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ FinUSGraph ↔ (𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2} ∧ 𝑉 ∈ Fin))) | ||
Theorem | isfusgrcl 29352 | The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 9-Jan-2020.) |
⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ (♯‘(Vtx‘𝐺)) ∈ ℕ0)) | ||
Theorem | fusgrusgr 29353 | A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) | ||
Theorem | opfusgr 29354 | A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.) |
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) | ||
Theorem | usgredgffibi 29355 | The number of edges in a simple graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 22-Oct-2020.) |
⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ Fin ↔ 𝐼 ∈ Fin)) | ||
Theorem | fusgredgfi 29356* | In a finite simple graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 21-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} ∈ Fin) | ||
Theorem | usgr1v0e 29357 | The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 1) → (♯‘𝐸) = 0) | ||
Theorem | usgrfilem 29358* | In a finite simple graph, the number of edges is finite iff the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝐸 ∈ Fin ↔ 𝐹 ∈ Fin)) | ||
Theorem | fusgrfisbase 29359 | Induction base for fusgrfis 29361. Main work is done in uhgr0v0e 29269. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.) |
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → 𝐸 ∈ Fin) | ||
Theorem | fusgrfisstep 29360* | Induction step in fusgrfis 29361: In a finite simple graph, the number of edges is finite if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.) |
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (( I ↾ {𝑝 ∈ (Edg‘〈𝑉, 𝐸〉) ∣ 𝑁 ∉ 𝑝}) ∈ Fin → 𝐸 ∈ Fin)) | ||
Theorem | fusgrfis 29361 | A finite simple graph is of finite size, i.e. has a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 8-Nov-2020.) |
⊢ (𝐺 ∈ FinUSGraph → (Edg‘𝐺) ∈ Fin) | ||
Theorem | fusgrfupgrfs 29362 | A finite simple graph is a finite pseudograph of finite size. (Contributed by AV, 27-Dec-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph → (𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin)) | ||
Syntax | cnbgr 29363 | Extend class notation with neighbors (of a vertex in a graph). |
class NeighbVtx | ||
Definition | df-nbgr 29364* |
Define the (open) neighborhood resp. the class of all neighbors of a
vertex (in a graph), see definition in section I.1 of [Bollobas] p. 3 or
definition in section 1.1 of [Diestel]
p. 3. The neighborhood/neighbors
of a vertex are all (other) vertices which are connected with this
vertex by an edge. In contrast to a closed neighborhood (see
df-clnbgr 47743), a vertex is not a neighbor of itself (see
nbgrnself 29390).
This definition is applicable even for arbitrary hypergraphs.
Remark: To distinguish this definition from other definitions for neighborhoods resp. neighbors (e.g., nei in Topology, see df-nei 23121), the suffix Vtx is added to the class constant NeighbVtx. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) |
⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) | ||
Theorem | nbgrprc0 29365 | The set of neighbors is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 26-Oct-2020.) |
⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 NeighbVtx 𝑁) = ∅) | ||
Theorem | nbgrcl 29366 | If a class 𝑋 has at least one neighbor, this class must be a vertex. (Contributed by AV, 6-Jun-2021.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ 𝑉) | ||
Theorem | nbgrval 29367* | The set of neighbors of a vertex 𝑉 in a graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}) | ||
Theorem | dfnbgr2 29368* | Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) | ||
Theorem | dfnbgr3 29369* | Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 29367). (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) (Revised by AV, 21-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}) | ||
Theorem | nbgrnvtx0 29370 | If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑋 ∉ 𝑉 → (𝐺 NeighbVtx 𝑋) = ∅) | ||
Theorem | nbgrel 29371* | Characterization of a neighbor 𝑁 of a vertex 𝑋 in a graph 𝐺. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ≠ 𝑋 ∧ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)) | ||
Theorem | nbgrisvtx 29372 | Every neighbor 𝑁 of a vertex 𝐾 is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁 ∈ 𝑉) | ||
Theorem | nbgrssvtx 29373 | The neighbors of a vertex 𝐾 in a graph form a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 NeighbVtx 𝐾) ⊆ 𝑉 | ||
Theorem | nbuhgr 29374* | The set of neighbors of a vertex in a hypergraph. This version of nbgrval 29367 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}) | ||
Theorem | nbupgr 29375* | The set of neighbors of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸}) | ||
Theorem | nbupgrel 29376 | A neighbor of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑁 ∈ 𝑉 ∧ 𝑁 ≠ 𝐾)) → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸)) | ||
Theorem | nbumgrvtx 29377* | The set of neighbors of a vertex in a multigraph. (Contributed by AV, 27-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) | ||
Theorem | nbumgr 29378* | The set of neighbors of an arbitrary class in a multigraph. (Contributed by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ UMGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) | ||
Theorem | nbusgrvtx 29379* | The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) | ||
Theorem | nbusgr 29380* | The set of neighbors of an arbitrary class in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}) | ||
Theorem | nbgr2vtx1edg 29381* | If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Revised by AV, 25-Mar-2021.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((♯‘𝑉) = 2 ∧ 𝑉 ∈ 𝐸) → ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)) | ||
Theorem | nbuhgr2vtx1edgblem 29382* | Lemma for nbuhgr2vtx1edgb 29383. This reverse direction of nbgr2vtx1edg 29381 only holds for classes whose edges are subsets of the set of vertices, which is the property of hypergraphs. (Contributed by AV, 2-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = {𝑎, 𝑏} ∧ 𝑎 ∈ (𝐺 NeighbVtx 𝑏)) → {𝑎, 𝑏} ∈ 𝐸) | ||
Theorem | nbuhgr2vtx1edgb 29383* | If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 2) → (𝑉 ∈ 𝐸 ↔ ∀𝑣 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣))) | ||
Theorem | nbusgreledg 29384 | A class/vertex is a neighbor of another class/vertex in a simple graph iff the vertices are endpoints of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑁, 𝐾} ∈ 𝐸)) | ||
Theorem | uhgrnbgr0nb 29385* | A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
⊢ ((𝐺 ∈ UHGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑁 ∉ 𝑒) → (𝐺 NeighbVtx 𝑁) = ∅) | ||
Theorem | nbgr0vtx 29386 | In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) (Proof shortened by AV, 10-May-2025.) |
⊢ ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) | ||
Theorem | nbgr0edglem 29387* | Lemma for nbgr0edg 29388 and nbgr1vtx 29389. (Contributed by AV, 15-Nov-2020.) |
⊢ (𝜑 → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒) ⇒ ⊢ (𝜑 → (𝐺 NeighbVtx 𝐾) = ∅) | ||
Theorem | nbgr0edg 29388 | In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.) |
⊢ ((Edg‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅) | ||
Theorem | nbgr1vtx 29389 | In a graph with one vertex, all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) |
⊢ ((♯‘(Vtx‘𝐺)) = 1 → (𝐺 NeighbVtx 𝐾) = ∅) | ||
Theorem | nbgrnself 29390* | A vertex in a graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ∀𝑣 ∈ 𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣) | ||
Theorem | nbgrnself2 29391 | A class 𝑋 is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑋 ∉ (𝐺 NeighbVtx 𝑋) | ||
Theorem | nbgrssovtx 29392 | The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 29373. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋}) | ||
Theorem | nbgrssvwo2 29393 | The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself and a class 𝑀 which is not a neighbor of 𝑋. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋})) | ||
Theorem | nbgrsym 29394 | In a graph, the neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Revised by AV, 12-Feb-2022.) |
⊢ (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁)) | ||
Theorem | nbupgrres 29395* | The neighborhood of a vertex in a restricted pseudograph (not necessarily valid for a hypergraph, because 𝑁, 𝐾 and 𝑀 could be connected by one edge, so 𝑀 is a neighbor of 𝐾 in the original graph, but not in the restricted graph, because the edge between 𝑀 and 𝐾, also incident with 𝑁, was removed). (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} & ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))) | ||
Theorem | usgrnbcnvfv 29396 | Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.) |
⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → (𝐼‘(◡𝐼‘{𝐾, 𝑁})) = {𝐾, 𝑁}) | ||
Theorem | nbusgredgeu 29397* | For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝑁)) → ∃!𝑒 ∈ 𝐸 𝑒 = {𝑀, 𝑁}) | ||
Theorem | edgnbusgreu 29398* | For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.) (Revised by AV, 6-Jul-2022.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑀) ⇒ ⊢ (((𝐺 ∈ USGraph ∧ 𝑀 ∈ 𝑉) ∧ (𝐶 ∈ 𝐸 ∧ 𝑀 ∈ 𝐶)) → ∃!𝑛 ∈ 𝑁 𝐶 = {𝑀, 𝑛}) | ||
Theorem | nbusgredgeu0 29399* | For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 13-Feb-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈) & ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ⇒ ⊢ (((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 𝑖 = {𝑈, 𝑀}) | ||
Theorem | nbusgrf1o0 29400* | The mapping of neighbors of a vertex to edges incident to the vertex is a bijection ( 1-1 onto function) in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑈) & ⊢ 𝐼 = {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} & ⊢ 𝐹 = (𝑛 ∈ 𝑁 ↦ {𝑈, 𝑛}) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → 𝐹:𝑁–1-1-onto→𝐼) |
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