A characterization of graphs G with G congruent to K-2(G)

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 D-1-graph if it has the T-1 property: for any two distinct vertices x and y of G, there exist cliques C and D of G such that x is an element of C but y is not an element of C and y is an element of D but x is not an element of D. Lim proved that if G is a D-1-graph, then G congruent to = K-2(G). Motivated by this result of Lim, we ask the following question: Can one characterize those graphs G with G congruent to K-2(G)? In this paper, we prove that in the class of D-graphs, G congruent to K-2(G) if and only if G has the T-1 property.