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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | usgredg2vlem1 27601* | Lemma 1 for usgredg2v 27603. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉) | ||
Theorem | usgredg2vlem2 27602* | Lemma 2 for usgredg2v 27603. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁})) | ||
Theorem | usgredg2v 27603* | In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} & ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ (℩𝑧 ∈ 𝑉 (𝐸‘𝑦) = {𝑧, 𝑁})) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1→𝑉) | ||
Theorem | usgriedgleord 27604* | Alternate version of usgredgleord 27609, not using the notation (Edg‘𝐺). In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ≤ (♯‘𝑉)) | ||
Theorem | ushgredgedg 27605* | In a simple hypergraph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 11-Dec-2020.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} & ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) ⇒ ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | usgredgedg 27606* | In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} & ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | ushgredgedgloop 27607* | In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex 𝑁 and the set of loops at this vertex 𝑁. (Contributed by AV, 11-Dec-2020.) (Revised by AV, 6-Jul-2022.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ (𝐼‘𝑖) = {𝑁}} & ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑒 = {𝑁}} & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) ⇒ ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | uspgredgleord 27608* | In a simple pseudograph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉)) | ||
Theorem | usgredgleord 27609* | In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉)) | ||
Theorem | usgredgleordALT 27610* | Alternate proof for usgredgleord 27609 based on usgriedgleord 27604. In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) (Proof shortened by AV, 5-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → (♯‘{𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒}) ≤ (♯‘𝑉)) | ||
Theorem | usgrstrrepe 27611* | Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened 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–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}) ⇒ ⊢ (𝜑 → (𝐺 sSet 〈𝐼, 𝐸〉) ∈ USGraph) | ||
Theorem | usgr0e 27612 | The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Proof shortened by AV, 25-Nov-2020.) |
⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → (iEdg‘𝐺) = ∅) ⇒ ⊢ (𝜑 → 𝐺 ∈ USGraph) | ||
Theorem | usgr0vb 27613 | The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 16-Oct-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) | ||
Theorem | uhgr0v0e 27614 | The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 = ∅) → 𝐸 = ∅) | ||
Theorem | uhgr0vsize0 27615 | The size of a hypergraph with no vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 0) → (♯‘𝐸) = 0) | ||
Theorem | uhgr0edgfi 27616 | A graph of order 0 (i.e. with 0 vertices) has a finite set of edges. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 8-Jun-2021.) |
⊢ ((𝐺 ∈ UHGraph ∧ (♯‘(Vtx‘𝐺)) = 0) → (Edg‘𝐺) ∈ Fin) | ||
Theorem | usgr0v 27617 | The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ USGraph) | ||
Theorem | uhgr0vusgr 27618 | The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.) |
⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → 𝐺 ∈ USGraph) | ||
Theorem | usgr0 27619 | The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.) |
⊢ ∅ ∈ USGraph | ||
Theorem | uspgr1e 27620 | A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (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 ∈ USPGraph) | ||
Theorem | usgr1e 27621 | A simple graph with one edge (with additional assumption that 𝐵 ≠ 𝐶 since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐶}〉}) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐺 ∈ USGraph) | ||
Theorem | usgr0eop 27622 | The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) |
⊢ (𝑉 ∈ 𝑊 → 〈𝑉, ∅〉 ∈ USGraph) | ||
Theorem | uspgr1eop 27623 | A simple pseudograph with (at least) two vertices and one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) |
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ USPGraph) | ||
Theorem | uspgr1ewop 27624 | A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 〈𝑉, 〈“{𝐴, 𝐵}”〉〉 ∈ USPGraph) | ||
Theorem | uspgr1v1eop 27625 | A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.) |
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑉) → 〈𝑉, {〈𝐴, {𝐵}〉}〉 ∈ USPGraph) | ||
Theorem | usgr1eop 27626 | A simple graph with (at least) two different vertices and one edge. If the two vertices were not different, the edge would be a loop. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.) |
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 ≠ 𝐶 → 〈𝑉, {〈𝐴, {𝐵, 𝐶}〉}〉 ∈ USGraph)) | ||
Theorem | uspgr2v1e2w 27627 | A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → 〈{𝐴, 𝐵}, 〈“{𝐴, 𝐵}”〉〉 ∈ USPGraph) | ||
Theorem | usgr2v1e2w 27628 | A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐴 ≠ 𝐵) → 〈{𝐴, 𝐵}, 〈“{𝐴, 𝐵}”〉〉 ∈ USGraph) | ||
Theorem | edg0usgr 27629 | A class without edges is a simple graph. Since ran 𝐹 = ∅ does not generally imply Fun 𝐹, but Fun (iEdg‘𝐺) is required for 𝐺 to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Edg‘𝐺) = ∅ ∧ Fun (iEdg‘𝐺)) → 𝐺 ∈ USGraph) | ||
Theorem | lfuhgr1v0e 27630* | A loop-free hypergraph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 2-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ (♯‘𝑉) = 1 ∧ 𝐼:dom 𝐼⟶𝐸) → (Edg‘𝐺) = ∅) | ||
Theorem | usgr1vr 27631 | A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 2-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑋 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph → (iEdg‘𝐺) = ∅)) | ||
Theorem | usgr1v 27632 | A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴}) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺) = ∅)) | ||
Theorem | usgr1v0edg 27633 | A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.) |
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = {𝐴} ∧ Fun (iEdg‘𝐺)) → (𝐺 ∈ USGraph ↔ (Edg‘𝐺) = ∅)) | ||
Theorem | usgrexmpldifpr 27634 | Lemma for usgrexmpledg 27638: all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.) |
⊢ (({0, 1} ≠ {1, 2} ∧ {0, 1} ≠ {2, 0} ∧ {0, 1} ≠ {0, 3}) ∧ ({1, 2} ≠ {2, 0} ∧ {1, 2} ≠ {0, 3} ∧ {2, 0} ≠ {0, 3})) | ||
Theorem | usgrexmplef 27635* | Lemma for usgrexmpl 27639. (Contributed by Alexander van der Vekens, 15-Aug-2017.) |
⊢ 𝑉 = (0...4) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 ⇒ ⊢ 𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2} | ||
Theorem | usgrexmpllem 27636 | Lemma for usgrexmpl 27639. (Contributed by AV, 21-Oct-2020.) |
⊢ 𝑉 = (0...4) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ ((Vtx‘𝐺) = 𝑉 ∧ (iEdg‘𝐺) = 𝐸) | ||
Theorem | usgrexmplvtx 27637 | The vertices 0, 1, 2, 3, 4 of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
⊢ 𝑉 = (0...4) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4}) | ||
Theorem | usgrexmpledg 27638 | The edges {0, 1}, {1, 2}, {2, 0}, {0, 3} of the graph 𝐺 = 〈𝑉, 𝐸〉. (Contributed by AV, 12-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
⊢ 𝑉 = (0...4) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ (Edg‘𝐺) = ({{0, 1}, {1, 2}} ∪ {{2, 0}, {0, 3}}) | ||
Theorem | usgrexmpl 27639 | 𝐺 is a simple graph of five vertices 0, 1, 2, 3, 4, with edges {0, 1}, {1, 2}, {2, 0}, {0, 3}. (Contributed by Alexander van der Vekens, 15-Aug-2017.) (Revised by AV, 21-Oct-2020.) |
⊢ 𝑉 = (0...4) & ⊢ 𝐸 = 〈“{0, 1} {1, 2} {2, 0} {0, 3}”〉 & ⊢ 𝐺 = 〈𝑉, 𝐸〉 ⇒ ⊢ 𝐺 ∈ USGraph | ||
Theorem | griedg0prc 27640* | The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.) |
⊢ 𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ⇒ ⊢ 𝑈 ∉ V | ||
Theorem | griedg0ssusgr 27641* | The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.) |
⊢ 𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ⇒ ⊢ 𝑈 ⊆ USGraph | ||
Theorem | usgrprc 27642 | The class of simple graphs is a proper class (and therefore, because of prcssprc 5250, the classes of multigraphs, pseudographs and hypergraphs are proper classes, too). (Contributed by AV, 27-Dec-2020.) |
⊢ USGraph ∉ V | ||
Syntax | csubgr 27643 | Extend class notation with subgraphs. |
class SubGraph | ||
Definition | df-subgr 27644* | Define the class of the subgraph relation. A class 𝑠 is a subgraph of a class 𝑔 (the supergraph of 𝑠) if its vertices are also vertices of 𝑔, and its edges are also edges of 𝑔, connecting vertices of 𝑠 only (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). The second condition is ensured by the requirement that the edge function of 𝑠 is a restriction of the edge function of 𝑔 having only vertices of 𝑠 in its range. Note that the domains of the edge functions of the subgraph and the supergraph should be compatible. (Contributed by AV, 16-Nov-2020.) |
⊢ SubGraph = {〈𝑠, 𝑔〉 ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))} | ||
Theorem | relsubgr 27645 | The class of the subgraph relation is a relation. (Contributed by AV, 16-Nov-2020.) |
⊢ Rel SubGraph | ||
Theorem | subgrv 27646 | 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 27647 | 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 27648 | 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 27649 | 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 27650 | 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 27651 | 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 27652 | 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 27653 | 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 27654 | 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 27655 | 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 27656 | 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 27657 | 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 27658 | 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 27659 | 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 27660 | 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 27661* | 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 27662 | 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 27663 | 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 27664 | A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.) |
⊢ ((𝐺 ∈ UMGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UMGraph) | ||
Theorem | subusgr 27665 | 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 27666 | Lemma for uhgrspansubgr 27667: 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 27667. (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 27667 | 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 27668 | 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 27669 | 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 27670 | 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 27671 | 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 27672 | 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 27673 | 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 27674 | 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 27675 | 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 27676 | Lemma 1 for uhgrspan1 27679. (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 27677 | Lemma 2 for uhgrspan1 27679. (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 27678 | Lemma 3 for uhgrspan1 27679. (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 27679* | 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 27680* | Lemma for upgrres 27682. (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 27681* | Lemma for umgrres 27683 and usgrres 27684. (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 27682* | A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 27679) 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 27683* | A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 27679) 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 27684* | A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 27679) 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 27685* | Lemma 1 for upgrres1 27689. (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 27686* | Lemma for umgrres1 27690. (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 27687* | Lemma 2 for upgrres1 27689. (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 27688* | Lemma 3 for upgrres1 27689. (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 27689* | 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 27644 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 27690* | 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 27644 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 27691* | 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 27644 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 27692 | Extend class notation with finite simple graphs. |
class FinUSGraph | ||
Definition | df-fusgr 27693 | 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 27694 | 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 27695 | 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 27696* | 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 27697 | 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 27698 | 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 27699 | 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 27700 | 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)) |
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