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Theorem List for Metamath Proof Explorer - 47801-47900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-grim 47801* An isomorphism between two graphs is a bijection between the sets of vertices of the two graphs that preserves adjacency, see definition in [Diestel] p. 3. (Contributed by AV, 19-Apr-2025.)
GraphIso = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘) / 𝑑](𝑗:dom 𝑒1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗𝑖)) = (𝑓 “ (𝑒𝑖))))})
 
Theoremgrimfn 47802 The graph isomorphism function is a well-defined function. (Contributed by AV, 28-Apr-2025.)
GraphIso Fn (V × V)
 
Theoremgrimdmrel 47803 The domain of the graph isomorphism function is a relation. (Contributed by AV, 28-Apr-2025.)
Rel dom GraphIso
 
Definitiondf-gric 47804 Two graphs are said to be isomorphic iff they are connected by at least one isomorphism, see definition in [Diestel] p. 3 and definition in [Bollobas] p. 3. Isomorphic graphs share all global graph properties like order and size. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 19-Apr-2025.)
𝑔𝑟 = ( GraphIso “ (V ∖ 1o))
 
Theoremisgrim 47805* An isomorphism of graphs is a bijection between their vertices that preserves adjacency. (Contributed by AV, 19-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (iEdg‘𝐺)    &   𝐷 = (iEdg‘𝐻)       ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∃𝑗(𝑗:dom 𝐸1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗𝑖)) = (𝐹 “ (𝐸𝑖))))))
 
Theoremgrimprop 47806* Properties of an isomorphism of graphs. (Contributed by AV, 29-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (iEdg‘𝐺)    &   𝐷 = (iEdg‘𝐻)       (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∃𝑗(𝑗:dom 𝐸1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗𝑖)) = (𝐹 “ (𝐸𝑖)))))
 
Theoremgrimf1o 47807 An isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 29-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)       (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉1-1-onto𝑊)
 
Theoremisuspgrim0lem 47808* An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑀 = (𝑥𝐸 ↦ (𝐹𝑥))    &   𝑁 = (𝑥 ∈ dom 𝐼 ↦ (𝐽‘(𝑀‘(𝐼𝑥))))       ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑀:𝐸1-1-onto𝐷) → (𝑁:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑁𝑖)) = (𝐹 “ (𝐼𝑖))))
 
Theoremisuspgrim0 47809* An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)       ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)))
 
Theoremuspgrimprop 47810* An isomorphism of simple pseudographs is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. (Contributed by AV, 27-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)       ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷))))
 
Theoremisuspgrimlem 47811* Lemma for isuspgrim 47812. (Contributed by AV, 27-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)       ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)
 
Theoremisuspgrim 47812* A class is an isomorphism of simple pseudographs iff it is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. This corresponds to the formal definition in [Bollobas] p. 3 and the definition in [Diestel] p. 3. (Contributed by AV, 27-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)       ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷))))
 
Theoremgrimidvtxedg 47813 The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and a graph with the same vertices and edges. (Contributed by AV, 4-May-2025.)
(𝜑𝐺 ∈ UHGraph)    &   (𝜑𝐻𝑉)    &   (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))    &   (𝜑 → (iEdg‘𝐺) = (iEdg‘𝐻))       (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))
 
Theoremgrimid 47814 The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and itself. (Contributed by AV, 29-Apr-2025.) (Prove shortened by AV, 5-May-2025.)
(𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))
 
Theoremgrimuhgr 47815 If there is a graph isomorphism between a hypergraph and a class with an edge function, the class is also a hypergraph. (Contributed by AV, 2-May-2025.)
((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇) ∧ Fun (iEdg‘𝑇)) → 𝑇 ∈ UHGraph)
 
Theoremgrimcnv 47816 The converse of a graph isomorphism is a graph isomorphism. (Contributed by AV, 1-May-2025.)
(𝑆 ∈ UHGraph → (𝐹 ∈ (𝑆 GraphIso 𝑇) → 𝐹 ∈ (𝑇 GraphIso 𝑆)))
 
Theoremgrimco 47817 The composition of graph isomorphisms is a graph isomorphism. (Contributed by AV, 3-May-2025.)
((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹𝐺) ∈ (𝑆 GraphIso 𝑈))
 
Theorembrgric 47818 The relation "is isomorphic to" for graphs. (Contributed by AV, 28-Apr-2025.)
(𝑅𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
 
Theorembrgrici 47819 Prove that two graphs are isomorphic by an explicit isomorphism. (Contributed by AV, 28-Apr-2025.)
(𝐹 ∈ (𝑅 GraphIso 𝑆) → 𝑅𝑔𝑟 𝑆)
 
Theoremgricrcl 47820 Reverse closure of the "is isomorphic to" relation for graphs. (Contributed by AV, 12-Jun-2025.)
(𝐺𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
 
Theoremdfgric2 47821* Alternate, explicit definition of the "is isomorphic to" relation for two graphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐼 = (iEdg‘𝐴)    &   𝐽 = (iEdg‘𝐵)       ((𝐴𝑋𝐵𝑌) → (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
 
Theoremgricbri 47822* Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.) (Proof shortened by AV, 12-Jun-2025.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐼 = (iEdg‘𝐴)    &   𝐽 = (iEdg‘𝐵)       (𝐴𝑔𝑟 𝐵 → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))
 
Theoremgricushgr 47823* The "is isomorphic to" relation for two simple hypergraphs. (Contributed by AV, 28-Nov-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)       ((𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph) → (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)))))
 
Theoremgricuspgr 47824* The "is isomorphic to" relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.) (Proof shortened by AV, 5-May-2025.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)       ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾))))
 
Theoremgricrel 47825 The "is isomorphic to" relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
Rel ≃𝑔𝑟
 
Theoremgricref 47826 Graph isomorphism is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 29-Apr-2025.)
(𝐺 ∈ UHGraph → 𝐺𝑔𝑟 𝐺)
 
Theoremgricsym 47827 Graph isomorphism is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 3-May-2025.)
(𝐺 ∈ UHGraph → (𝐺𝑔𝑟 𝑆𝑆𝑔𝑟 𝐺))
 
Theoremgricsymb 47828 Graph isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Proof shortened by AV, 3-May-2025.)
((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴𝑔𝑟 𝐵𝐵𝑔𝑟 𝐴))
 
Theoremgrictr 47829 Graph isomorphism is transitive. (Contributed by AV, 5-Dec-2022.) (Revised by AV, 3-May-2025.)
((𝑅𝑔𝑟 𝑆𝑆𝑔𝑟 𝑇) → 𝑅𝑔𝑟 𝑇)
 
Theoremgricer 47830 Isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 3-May-2025.) (Proof shortened by AV, 11-Jul-2025.)
( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
 
Theoremgricen 47831 Isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 3-May-2025.)
𝐵 = (Vtx‘𝑅)    &   𝐶 = (Vtx‘𝑆)       (𝑅𝑔𝑟 𝑆𝐵𝐶)
 
Theoremopstrgric 47832 A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 4-May-2025.)
𝐺 = ⟨𝑉, 𝐸    &   𝐻 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}       ((𝐺 ∈ UHGraph ∧ 𝑉𝑋𝐸𝑌) → 𝐺𝑔𝑟 𝐻)
 
Theoremushggricedg 47833 A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐻 = ⟨𝑉, ( I ↾ 𝐸)⟩       (𝐺 ∈ USHGraph → 𝐺𝑔𝑟 𝐻)
 
Theoremisubgrgrim 47834* Isomorphic subgraphs induced by subsets of vertices of two graphs. (Contributed by AV, 29-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}       (((𝐺𝑈𝐻𝑇) ∧ (𝑁𝑉𝑀𝑊)) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
 
Theoremuhgrimisgrgriclem 47835* Lemma for uhgrimisgrgric 47836. (Contributed by AV, 31-May-2025.)
(((𝐹:𝑉1-1-onto𝑊𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁𝑉𝐼:𝐴1-1-onto𝐵) ∧ ∀𝑖𝐴 (𝐻‘(𝐼𝑖)) = (𝐹 “ (𝐺𝑖))) → ((𝐽𝐵 ∧ (𝐻𝐽) ⊆ (𝐹𝑁)) ↔ ∃𝑘𝐴 ((𝐺𝑘) ⊆ 𝑁 ∧ (𝐼𝑘) = 𝐽)))
 
Theoremuhgrimisgrgric 47836 For isomorphic hypergraphs, the induced subgraph of a subset of vertices of one graph is isomorphic to the subgraph induced by the image of the subset. (Contributed by AV, 31-May-2025.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝑁𝑉) → (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐹𝑁)))
 
Theoremclnbgrisubgrgrim 47837* Isomorphic subgraphs induced by closed neighborhoods of vertices of two graphs. (Contributed by AV, 29-May-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑋)    &   𝑀 = (𝐻 ClNeighbVtx 𝑌)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}       ((𝐺𝑈𝐻𝑇) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
 
Theoremclnbgrgrimlem 47838* Lemma for clnbgrgrim 47839: For two isomorphic hypergraphs, if there is an edge connecting the image of a vertex of the first graph with a vertex of the second graph, the vertex of the second graph is the image of a neighbor of the vertex of the first graph. (Contributed by AV, 2-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐻)       (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝑋𝑉𝑌𝑊)) → ((𝐾𝐸 ∧ {(𝐹𝑋), 𝑌} ⊆ 𝐾) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹𝑛) = 𝑌))
 
Theoremclnbgrgrim 47839 Graph isomorphisms between hypergraphs map closed neighborhoods onto closed neighborhoods. (Contributed by AV, 2-Jun-2025.)
𝑉 = (Vtx‘𝐺)       ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋𝑉) → (𝐻 ClNeighbVtx (𝐹𝑋)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑋)))
 
Theoremgrimedg 47840 Graph isomorphisms map edges onto the corresponding edges. (Contributed by AV, 7-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (Edg‘𝐺)    &   𝐸 = (Edg‘𝐻)       ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾𝐼 ↔ ((𝐹𝐾) ∈ 𝐸𝐾𝑉)))
 
21.48.15.5  Triangles in graphs

Usually, a "triangle" in graph theory is a complete graph consisting of three vertices (denoted by " K3 "), see the definition in [Diestel] p. 3 or the definition in [Bollobas] p. 5. This corresponds to the definition of a "triangle graph" (which is a more precise term) in Wikipedia "Triangle graph", https://en.wikipedia.org/wiki/Triangle_graph, 27-Jul-2025: "In the mathematical field of graph theory, the triangle graph is a planar undirected graph with 3 vertices and 3 edges, in the form of a triangle. The triangle graph is also known as the cycle graph C3 and the complete graph K3."

Often, however, the term "triangle" is also used to denote a corresponding subgraph of a given graph ("triangle in a graph"), see, for example, Wikipedia "Triangle-free graph", 28-Jul-2025, https://en.wikipedia.org/wiki/Triangle-free_graph: "In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges."

In this subsection, a triangle (in a graph) is defined as a set of three vertices of a given graph. In this meaning, a triangle 𝑇 with (𝑇 ∈ (GrTriangles‘𝐺)) is neither a graph nor a subgraph, but it induces a triangle graph (𝐺 ISubGr 𝑇) as subgraph of the given graph 𝐺.

We require that there are three (different) edges connecting the three (different) vertices of the triangle. Therefore, it is not sufficient for arbitrary hypergraphs to say "a triangle is a set of three (different) vertices connected with each other (by edges)", because there might be only one or two multiedges fulfilling this statement. We do not regard such degenerate cases as "triangle".

The definition df-grtri 47842 is designed for a special purpose, namely to provide a criterion for two graphs being not isomorphic (see grimgrtri 47851). For other purposes, a more general definition might be useful, e.g., ComplSubGr = (𝑔 ∈ V, 𝑛 ∈ ℕ ↦ {𝑡 ∈ 𝒫 𝑣 ∣ ((♯‘𝑡) = 𝑛 ∧ (𝑔 ISubGr 𝑡) ∈ ComplGraph)}) for complete subgraphs of a given size (proposed by TA). With such a definition, we would have (GrTriangles‘𝐺) = (𝐺 ComplSubGr 3) (at least for simple graphs), and the definition df-grtri 47842 may become obsolete.

 
Syntaxcgrtri 47841 Extend class notation with triangles (in a graph).
class GrTriangles
 
Definitiondf-grtri 47842* Definition of a triangles in a graph. A triangle in a graph is a set of three (different) vertices completely connected with each other. Such vertices induce a closed walk of length 3, see grtriclwlk3 47849. (TODO: and a cycle of length 3 ,see grtricycl ). (Contributed by AV, 20-Jul-2025.)
GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
 
Theoremgrtriproplem 47843 Lemma for grtriprop 47845. (Contributed by AV, 23-Jul-2025.)
((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
 
Theoremgrtri 47844* The triangles in a graph. (Contributed by AV, 20-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
 
Theoremgrtriprop 47845* The properties of a triangle. (Contributed by AV, 25-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
 
Theoremgrtrif1o 47846 Any bijection onto a triangle preserves the edges of the triangle. (Contributed by AV, 25-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝐹:(0..^3)–1-1-onto𝑇) → ({(𝐹‘0), (𝐹‘1)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹‘2)} ∈ 𝐸 ∧ {(𝐹‘1), (𝐹‘2)} ∈ 𝐸))
 
Theoremisgrtri 47847* A triangle in a graph. (Contributed by AV, 20-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑇 ∈ (GrTriangles‘𝐺) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
 
Theoremgrtrissvtx 47848 A triangle is a subset of the vertices (of a graph). (Contributed by AV, 26-Jul-2025.)
𝑉 = (Vtx‘𝐺)       (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇𝑉)
 
Theoremgrtriclwlk3 47849 A triangle induces a closed walk of length 3 . (Contributed by AV, 26-Jul-2025.)
(𝜑𝑇 ∈ (GrTriangles‘𝐺))    &   (𝜑𝑃:(0..^3)–1-1-onto𝑇)       (𝜑𝑃 ∈ (3 ClWWalksN 𝐺))
 
Theoremgrtrimap 47850 Conditions for mapping triangles onto triangles. Lemma for grimgrtri 47851 and grlimgrtri 47898. (Contributed by AV, 23-Aug-2025.)
(𝐹:𝑉1-1𝑊 → (((𝑎𝑉𝑏𝑉𝑐𝑉) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3)) → (((𝐹𝑎) ∈ 𝑊 ∧ (𝐹𝑏) ∈ 𝑊 ∧ (𝐹𝑐) ∈ 𝑊) ∧ (𝐹𝑇) = {(𝐹𝑎), (𝐹𝑏), (𝐹𝑐)} ∧ (♯‘(𝐹𝑇)) = 3)))
 
Theoremgrimgrtri 47851 Graph isomorphisms map triangles onto triangles. (Contributed by AV, 27-Jul-2025.) (Proof shortened by AV, 24-Aug-2025.)
(𝜑𝐺 ∈ UHGraph)    &   (𝜑𝐻 ∈ UHGraph)    &   (𝜑𝐹 ∈ (𝐺 GraphIso 𝐻))    &   (𝜑𝑇 ∈ (GrTriangles‘𝐺))       (𝜑 → (𝐹𝑇) ∈ (GrTriangles‘𝐻))
 
Theoremusgrgrtrirex 47852* Conditions for a simple graph to contain a triangle. (Contributed by AV, 7-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑎)       (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎𝑉𝑏𝑁𝑐𝑁 (𝑏𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
 
21.48.15.6  Star graphs

According to Wikipedia "Star (graph theory)", 10-Sep-2025, https://en.wikipedia.org/wiki/Star_(graph_theory): "In graph theory, the star Sk is the complete bipartite graph K(1,k), that is, it is a tree with one internal node and k leaves. Alternatively, some authors define Sk to be the tree of order k with maximum diameter 2, in which case a star of k > 2 has k - 1 leaves.".

 
Syntaxcstgr 47853 Extend class notation with star graphs.
class StarGr
 
Definitiondf-stgr 47854* Definition of star graphs according to the first definition in Wikipedia, so that (StarGr‘𝑁) has size 𝑁, and order 𝑁 + 1: (StarGr‘0) will be a single vertex (graph without edges), see stgr0 47862, (StarGr‘1) will be a single edge (graph with two vertices connected by an edge), see stgr1 47863, and (StarGr‘3) will be a 3-star or "claw" (a star with 3 edges). (Contributed by AV, 10-Sep-2025.)
StarGr = (𝑛 ∈ ℕ0 ↦ {⟨(Base‘ndx), (0...𝑛)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑛) ∣ ∃𝑥 ∈ (1...𝑛)𝑒 = {0, 𝑥}})⟩})
 
Theoremstgrfv 47855* The star graph SN. (Contributed by AV, 10-Sep-2025.)
(𝑁 ∈ ℕ0 → (StarGr‘𝑁) = {⟨(Base‘ndx), (0...𝑁)⟩, ⟨(.ef‘ndx), ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})⟩})
 
Theoremstgrvtx 47856 The vertices of the star graph SN. (Contributed by AV, 11-Sep-2025.)
(𝑁 ∈ ℕ0 → (Vtx‘(StarGr‘𝑁)) = (0...𝑁))
 
Theoremstgriedg 47857* The indexed edges of the star graph SN. (Contributed by AV, 11-Sep-2025.)
(𝑁 ∈ ℕ0 → (iEdg‘(StarGr‘𝑁)) = ( I ↾ {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}}))
 
Theoremstgredg 47858* The edges of the star graph SN. (Contributed by AV, 11-Sep-2025.)
(𝑁 ∈ ℕ0 → (Edg‘(StarGr‘𝑁)) = {𝑒 ∈ 𝒫 (0...𝑁) ∣ ∃𝑥 ∈ (1...𝑁)𝑒 = {0, 𝑥}})
 
Theoremstgredgel 47859* An edge of the star graph SN. (Contributed by AV, 11-Sep-2025.)
(𝑁 ∈ ℕ0 → (𝐸 ∈ (Edg‘(StarGr‘𝑁)) ↔ (𝐸 ⊆ (0...𝑁) ∧ ∃𝑥 ∈ (1...𝑁)𝐸 = {0, 𝑥})))
 
Theoremstgredgiun 47860* The edges of the star graph SN as indexed union. (Contributed by AV, 29-Sep-2025.)
(𝑁 ∈ ℕ0 → (Edg‘(StarGr‘𝑁)) = 𝑥 ∈ (1...𝑁){{0, 𝑥}})
 
Theoremstgrusgra 47861 The star graph SN is a simple graph. (Contributed by AV, 11-Sep-2025.)
(𝑁 ∈ ℕ0 → (StarGr‘𝑁) ∈ USGraph)
 
Theoremstgr0 47862 The star graph S0 consists of a single vertex without edges. (Contributed by AV, 11-Sep-2025.)
(StarGr‘0) = {⟨(Base‘ndx), {0}⟩, ⟨(.ef‘ndx), ∅⟩}
 
Theoremstgr1 47863 The star graph S1 consists of a single simple edge. (Contributed by AV, 11-Sep-2025.)
(StarGr‘1) = {⟨(Base‘ndx), {0, 1}⟩, ⟨(.ef‘ndx), ( I ↾ {{0, 1}})⟩}
 
Theoremstgrvtx0 47864 The center ("internal node") of a star graph SN. (Contributed by AV, 12-Sep-2025.)
𝐺 = (StarGr‘𝑁)    &   𝑉 = (Vtx‘𝐺)       (𝑁 ∈ ℕ0 → 0 ∈ 𝑉)
 
Theoremstgrorder 47865 The order of a star graph SN. (Contributed by AV, 12-Sep-2025.)
𝐺 = (StarGr‘𝑁)    &   𝑉 = (Vtx‘𝐺)       (𝑁 ∈ ℕ0 → (♯‘𝑉) = (𝑁 + 1))
 
Theoremstgrnbgr0 47866 All vertices of a star graph SN except the center are in the (open) neighborhood of the center. (Contributed by AV, 12-Sep-2025.)
𝐺 = (StarGr‘𝑁)    &   𝑉 = (Vtx‘𝐺)       (𝑁 ∈ ℕ0 → (𝐺 NeighbVtx 0) = (𝑉 ∖ {0}))
 
Theoremstgrclnbgr0 47867 All vertices of a star graph SN are in the closed neighborhood of the center. (Contributed by AV, 12-Sep-2025.)
𝐺 = (StarGr‘𝑁)    &   𝑉 = (Vtx‘𝐺)       (𝑁 ∈ ℕ0 → (𝐺 ClNeighbVtx 0) = 𝑉)
 
Theoremisubgr3stgrlem1 47868 Lemma 1 for isubgr3stgr 47877. (Contributed by AV, 16-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝐹 = (𝐻 ∪ {⟨𝑋, 𝑌⟩})       ((𝐻:𝑈1-1-onto𝑅𝑋𝑉 ∧ (𝑌𝑊𝑌𝑅)) → 𝐹:𝐶1-1-onto→(𝑅 ∪ {𝑌}))
 
Theoremisubgr3stgrlem2 47869* Lemma 2 for isubgr3stgr 47877. (Contributed by AV, 16-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)       ((𝐺 ∈ USGraph ∧ 𝑋𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑓 𝑓:𝑈1-1-onto→(𝑊 ∖ {0}))
 
Theoremisubgr3stgrlem3 47870* Lemma 3 for isubgr3stgr 47877. (Contributed by AV, 17-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)       ((𝐺 ∈ USGraph ∧ 𝑋𝑉 ∧ (♯‘𝑈) = 𝑁) → ∃𝑔(𝑔:𝐶1-1-onto𝑊 ∧ (𝑔𝑋) = 0))
 
Theoremisubgr3stgrlem4 47871* Lemma 4 for isubgr3stgr 47877. (Contributed by AV, 24-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)       ((𝐴 = 𝑋 ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ∃𝑧 ∈ (1...𝑁)(𝐹 “ {𝐴, 𝐵}) = {0, 𝑧})
 
Theoremisubgr3stgrlem5 47872* Lemma 5 for isubgr3stgr 47877. (Contributed by AV, 24-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))       ((𝐹:𝐶𝑊𝑌𝐼) → (𝐻𝑌) = (𝐹𝑌))
 
Theoremisubgr3stgrlem6 47873* Lemma 6 for isubgr3stgr 47877. (Contributed by AV, 24-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))       ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐻:𝐼⟶(Edg‘(StarGr‘𝑁)))
 
Theoremisubgr3stgrlem7 47874* Lemma 7 for isubgr3stgr 47877. (Contributed by AV, 29-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))       (((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0) ∧ 𝐽 ∈ (Edg‘(StarGr‘𝑁))) → (𝐹𝐽) ∈ 𝐼)
 
Theoremisubgr3stgrlem8 47875* Lemma 8 for isubgr3stgr 47877. (Contributed by AV, 29-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))       ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → 𝐻:𝐼1-1-onto→(Edg‘(StarGr‘𝑁)))
 
Theoremisubgr3stgrlem9 47876* Lemma 9 for isubgr3stgr 47877. (Contributed by AV, 29-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)    &   𝐼 = (Edg‘(𝐺 ISubGr 𝐶))    &   𝐻 = (𝑖𝐼 ↦ (𝐹𝑖))       ((((𝐺 ∈ USGraph ∧ 𝑋𝑉) ∧ ((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸)) ∧ (𝐹:𝐶1-1-onto𝑊 ∧ (𝐹𝑋) = 0)) → (𝐻:𝐼1-1-onto→(Edg‘(StarGr‘𝑁)) ∧ ∀𝑒𝐼 (𝐹𝑒) = (𝐻𝑒)))
 
Theoremisubgr3stgr 47877* If a vertex of a simple graph has exactly 𝑁 (different) neighbors, and none of these neighbors are connected by an edge, then the (closed) neighborhood of this vertex induces a subgraph which is isomorphic to an 𝑁-star. (Contributed by AV, 29-Sep-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑈 = (𝐺 NeighbVtx 𝑋)    &   𝐶 = (𝐺 ClNeighbVtx 𝑋)    &   𝑁 ∈ ℕ0    &   𝑆 = (StarGr‘𝑁)    &   𝑊 = (Vtx‘𝑆)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑋𝑉) → (((♯‘𝑈) = 𝑁 ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∉ 𝐸) → (𝐺 ISubGr 𝐶) ≃𝑔𝑟 (StarGr‘𝑁)))
 
21.48.15.7  Local isomorphisms of graphs

This section is about local isomorphisms of graphs, which are a generalization of isomorphisms of graphs, i.e., every isomorphism between two graphs is also a local isomorphism between these graphs, see uhgrimgrlim 47889.

This definition is according to a chat in mathoverflow (https://mathoverflow.net/questions/491133/locally-isomorphic-graphs 47889): roughly speaking, it restricts the correspondence of two graphs to their neighborhoods.

Additionally, a binary relation 𝑙𝑔𝑟 is defined (see df-grlic 47883) which is true for two graphs iff there is a local isomorphism between these graphs. Then these graphs are called "locally isomorphic". Therefore, this relation is also called "is locally isomorphic to" relation. As a main result of this section, it is shown that the "is locally isomorphic to" relation is an equivalence relation (for hypergraphs), see grlicer 47911.

The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19289) and graph isomorphisms GraphIso (see df-grim 47801) resp. isomorphism between groups 𝑔 (see df-gic 19290) and isomorphism between graphs 𝑔𝑟 (see df-gric 47804).

In the future, it should be shown that there are local isomorphisms between two graphs which are not (ordinary) isomorphisms between these graphs, as dicussed in the above mentioned chat in mathoverflow.

 
Syntaxcgrlim 47878 The class of graph local isomorphism sets.
class GraphLocIso
 
Syntaxcgrlic 47879 The class of the graph local isomorphism relation.
class 𝑙𝑔𝑟
 
Definitiondf-grlim 47880* A local isomorphism of graphs is a bijection between the sets of vertices of two graphs that preserves local adjacency, i.e. the subgraph induced by the closed neighborhood of a vertex of the first graph and the subgraph induced by the closed neighborhood of the associated vertex of the second graph are isomorphic. See the following chat in mathoverflow: https://mathoverflow.net/questions/491133/locally-isomorphic-graphs. (Contributed by AV, 27-Apr-2025.)
GraphLocIso = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))})
 
Theoremgrlimfn 47881 The graph local isomorphism function is a well-defined function. (Contributed by AV, 20-May-2025.)
GraphLocIso Fn (V × V)
 
Theoremgrlimdmrel 47882 The domain of the graph local isomorphism function is a relation. (Contributed by AV, 20-May-2025.)
Rel dom GraphLocIso
 
Definitiondf-grlic 47883 Two graphs are said to be locally isomorphic iff they are connected by at least one local isomorphism. (Contributed by AV, 27-Apr-2025.)
𝑙𝑔𝑟 = ( GraphLocIso “ (V ∖ 1o))
 
Theoremisgrlim 47884* A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. (Contributed by AV, 20-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)       ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
 
Theoremisgrlim2 47885* A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}       ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
 
Theoremgrlimprop 47886* Properties of a local isomorphism of graphs. (Contributed by AV, 21-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)       (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))))
 
Theoremgrlimf1o 47887 A local isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 21-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)       (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹:𝑉1-1-onto𝑊)
 
Theoremgrlimprop2 47888* Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}       (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
 
Theoremuhgrimgrlim 47889 An isomorphism of hypergraphs is a local isomorphism between the two graphs. (Contributed by AV, 2-Jun-2025.)
((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
 
Theoremuspgrlimlem1 47890* Lemma 1 for uspgrlim 47894. (Contributed by AV, 16-Aug-2025.)
𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥𝐽𝑥𝑀}       (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
 
Theoremuspgrlimlem2 47891* Lemma 2 for uspgrlim 47894. (Contributed by AV, 16-Aug-2025.)
𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥𝐽𝑥𝑀}       (𝐻 ∈ USPGraph → ((iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})
 
Theoremuspgrlimlem3 47892* Lemma 3 for uspgrlim 47894. (Contributed by AV, 16-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥𝐼𝑥𝑁}    &   𝐿 = {𝑥𝐽𝑥𝑀}       ((𝐺 ∈ USPGraph ∧ :{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(𝑖))) → (𝑒𝐾 → (𝑓𝑒) = ((((iEdg‘𝐻) ∘ ) ∘ (iEdg‘𝐺))‘𝑒)))
 
Theoremuspgrlimlem4 47893* Lemma 4 for uspgrlim 47894. (Contributed by AV, 16-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥𝐼𝑥𝑁}    &   𝐿 = {𝑥𝐽𝑥𝑀}       (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘((((iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))
 
Theoremuspgrlim 47894* A local isomorphism of simple pseudographs is a bijection between their vertices that preserves neighborhoods, expressed by properties of their edges (not edge functions as in isgrlim2 47885). (Contributed by AV, 15-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥𝐼𝑥𝑁}    &   𝐿 = {𝑥𝐽𝑥𝑀}       ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))))))
 
Theoremusgrlimprop 47895* Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥𝐼𝑥𝑁}    &   𝐿 = {𝑥𝐽𝑥𝑀}       ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒)))))
 
Theoremgrlimgrtrilem1 47896* Lemma 3 for grlimgrtri 47898. (Contributed by AV, 24-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥𝐼𝑥𝑁}       ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))
 
Theoremgrlimgrtrilem2 47897* Lemma 3 for grlimgrtri 47898. (Contributed by AV, 23-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥𝐼𝑥𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹𝑎))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥𝐽𝑥𝑀}       (((𝑓:𝑁1-1-onto𝑀𝑔:𝐾1-1-onto𝐿) ∧ ∀𝑖𝐾 (𝑓𝑖) = (𝑔𝑖) ∧ {𝑏, 𝑐} ∈ 𝐾) → {(𝑓𝑏), (𝑓𝑐)} ∈ 𝐽)
 
Theoremgrlimgrtri 47898* Local isomorphisms between simple pseudographs map triangles onto triangles. (Contributed by AV, 24-Aug-2025.)
(𝜑𝐺 ∈ USPGraph)    &   (𝜑𝐻 ∈ USPGraph)    &   (𝜑𝐹 ∈ (𝐺 GraphLocIso 𝐻))    &   (𝜑𝑇 ∈ (GrTriangles‘𝐺))       (𝜑 → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐻))
 
Theorembrgrlic 47899 The relation "is locally isomorphic to" for graphs. (Contributed by AV, 9-Jun-2025.)
(𝑅𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
 
Theorembrgrilci 47900 Prove that two graphs are locally isomorphic by an explicit local isomorphism. (Contributed by AV, 9-Jun-2025.)
(𝐹 ∈ (𝑅 GraphLocIso 𝑆) → 𝑅𝑙𝑔𝑟 𝑆)
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