On graphs whose complement and square are isomorphic Martin Milani - PowerPoint PPT Presentation

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