A characterization of graphs G with G ≈ K2(G) Original Research Article

A graph G is called a D-graph if for every set of cliques of G whose pairwise intersections are nonempty there is a vertex of G common to all the cliques of the set. A D-graph G is called a D1-graph if it has the T1 property: for any two distinct vertices x and y of G, there exist cliques C and D of G such that x ∈ C but y ∉ C and y ∈ D but x ∉ D. Lim proved that if G is a D1-graph, then G ≅ K2(G). Motivated by this result of Lim, we ask the following question: Can one characterize those graphs G with G ≅ K2(G)? In this paper, we prove that in the class of D-graphs, G ≅ K2(G) if and only if G has the T1 property.