On graphs whose complement and square are isomorphic Martin Milaniˇ c, Anders Sune Pedersen, Daniel Pellicer University of Primorska, Koper, Slovenia University of Southern Denmark UNAM, Mexico 26 th Ljubljana-Leoben Seminar, Bovec, Slovenia 20th September, 2012 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Main definitions Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graph complement G : a simple graph The complement of G is the graph G defined as: V ( G ) = V ( G ) E ( G ) = { uv : u , v ∈ V ( G ) ∧ u � = v ∧ uv �∈ E ( G ) } Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graph complement G : a simple graph The complement of G is the graph G defined as: V ( G ) = V ( G ) E ( G ) = { uv : u , v ∈ V ( G ) ∧ u � = v ∧ uv �∈ E ( G ) } Example: G Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graph complement G : a simple graph The complement of G is the graph G defined as: V ( G ) = V ( G ) E ( G ) = { uv : u , v ∈ V ( G ) ∧ u � = v ∧ uv �∈ E ( G ) } Example: G Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graph squares G : a simple graph The square of G is the graph G 2 defined as: V ( G 2 ) = V ( G ) E ( G 2 ) = { uv : 1 ≤ dist G ( u , v ) ≤ 2 } Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graph squares G : a simple graph The square of G is the graph G 2 defined as: V ( G 2 ) = V ( G ) E ( G 2 ) = { uv : 1 ≤ dist G ( u , v ) ≤ 2 } Example: G Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graph squares G : a simple graph The square of G is the graph G 2 defined as: V ( G 2 ) = V ( G ) E ( G 2 ) = { uv : 1 ≤ dist G ( u , v ) ≤ 2 } Example: G 2 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graphs whose complement is isomorphic to ... Recall that two graphs G and H are isomorphic if there exists an isomorphism from G to H , that is, a bijection ϕ : V ( G ) → V ( H ) that preserves adjacencies and non-adjacencies. Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graphs whose complement is isomorphic to ... Recall that two graphs G and H are isomorphic if there exists an isomorphism from G to H , that is, a bijection ϕ : V ( G ) → V ( H ) that preserves adjacencies and non-adjacencies. Research problem: Given a graph transformation ψ , characterize graphs G such that G is isomorphic to ψ ( G ) . Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graphs whose complement is isomorphic to ... Recall that two graphs G and H are isomorphic if there exists an isomorphism from G to H , that is, a bijection ϕ : V ( G ) → V ( H ) that preserves adjacencies and non-adjacencies. Research problem: Given a graph transformation ψ , characterize graphs G such that G is isomorphic to ψ ( G ) . ψ ( G ) = L ( G ) : Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graphs whose complement is isomorphic to ... Recall that two graphs G and H are isomorphic if there exists an isomorphism from G to H , that is, a bijection ϕ : V ( G ) → V ( H ) that preserves adjacencies and non-adjacencies. Research problem: Given a graph transformation ψ , characterize graphs G such that G is isomorphic to ψ ( G ) . ψ ( G ) = L ( G ) : There are only two graphs such that their complement is isomorphic to their line graph. (Martin Aigner (JCTB, 1969)) Martin Milaniˇ c On graphs whose complement and square are isomorphic
Graphs whose complement is isomorphic to ... Recall that two graphs G and H are isomorphic if there exists an isomorphism from G to H , that is, a bijection ϕ : V ( G ) → V ( H ) that preserves adjacencies and non-adjacencies. Research problem: Given a graph transformation ψ , characterize graphs G such that G is isomorphic to ψ ( G ) . ψ ( G ) = L ( G ) : There are only two graphs such that their complement is isomorphic to their line graph. (Martin Aigner (JCTB, 1969)) ψ ( G ) = G : self-complementary graphs Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs We are interested in the case ψ ( G ) = G 2 . Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs We are interested in the case ψ ( G ) = G 2 . Definition A graph G is said to be square-complementary (squco) if G 2 is isomorphic to G . Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs We are interested in the case ψ ( G ) = G 2 . Definition A graph G is said to be square-complementary (squco) if G 2 is isomorphic to G . Equivalently: G is isomorphic to G 2 G is isomorphic to G 2 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Examples Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Example: C 7 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Example: C 2 7 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Example: C 2 7 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another (bipartite) example: the Franklin graph F Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another (bipartite) example: the Franklin graph F Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another (bipartite) example: the Franklin graph F 2 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another (bipartite) example: the Franklin graph F 2 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another (bipartite) example: the Franklin graph 11 2 4 9 F 2 1 12 7 6 10 3 8 5 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another (bipartite) example: the Franklin graph 1 2 12 3 F 2 4 11 5 10 6 9 8 7 Martin Milaniˇ c On graphs whose complement and square are isomorphic
Square-complementary graphs Another bipartite example, on 13 vertices: Martin Milaniˇ c On graphs whose complement and square are isomorphic
Constructions Martin Milaniˇ c On graphs whose complement and square are isomorphic
Infinite families of squco graphs G : a graph, k : a positive integer G [ k ] : the graph obtained by Martin Milaniˇ c On graphs whose complement and square are isomorphic
Infinite families of squco graphs G : a graph, k : a positive integer G [ k ] : the graph obtained by replacing every vertex u with a set of k pairwise non-adjacent vertices S ( u ) and Martin Milaniˇ c On graphs whose complement and square are isomorphic
Infinite families of squco graphs G : a graph, k : a positive integer G [ k ] : the graph obtained by replacing every vertex u with a set of k pairwise non-adjacent vertices S ( u ) and connecting every vertex in S ( u ) with every vertex in S ( v ) if and only if u and v are adjacent in G . Martin Milaniˇ c On graphs whose complement and square are isomorphic
Infinite families of squco graphs G : a graph, k : a positive integer G [ k ] : the graph obtained by replacing every vertex u with a set of k pairwise non-adjacent vertices S ( u ) and connecting every vertex in S ( u ) with every vertex in S ( v ) if and only if u and v are adjacent in G . For every positive integer k , if G is a nontrivial squco graph, then also G [ k ] is squco. Martin Milaniˇ c On graphs whose complement and square are isomorphic
Infinite families of squco graphs G : a graph, k : a positive integer G [ k ] : the graph obtained by replacing every vertex u with a set of k pairwise non-adjacent vertices S ( u ) and connecting every vertex in S ( u ) with every vertex in S ( v ) if and only if u and v are adjacent in G . For every positive integer k , if G is a nontrivial squco graph, then also G [ k ] is squco. For every nontrivial vertex transitive squco graph G and every u ∈ V ( G ) , Martin Milaniˇ c On graphs whose complement and square are isomorphic
Infinite families of squco graphs G : a graph, k : a positive integer G [ k ] : the graph obtained by replacing every vertex u with a set of k pairwise non-adjacent vertices S ( u ) and connecting every vertex in S ( u ) with every vertex in S ( v ) if and only if u and v are adjacent in G . For every positive integer k , if G is a nontrivial squco graph, then also G [ k ] is squco. For every nontrivial vertex transitive squco graph G and every u ∈ V ( G ) , the graph obtained by replacing u with a set of k ≥ 1 non-adjacent vertices Martin Milaniˇ c On graphs whose complement and square are isomorphic
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