A hierarchy of self-clique graphs Original Research Article

The clique graph K(G) of G is the intersection graph of all its (maximal) cliques. A connected graph G is self-clique whenever G≅K(G). Self-clique graphs have been studied in several papers. Here we propose a hierarchy of self-clique graphs: Type 3 ⊊ Type 2 ⊊ Type 1 ⊊ Type 0. We give characterizations for classes of Types 3, 2 and 1 (including Helly self-clique graphs) and several new constructions of families of self-clique graphs. It is shown that all (but one) previously published examples of self-clique graphs are of Type 2. Our methods provide a unified approach and generalizations of those examples. As further applications, we give a characterization of the self-clique graphs such that at most 3 cliques have more than two vertices (they are all of Type 2) and a description of the diamond-free graphs of Type 2.