On clique-complete graphs Original Research Article

The clique graph, K(G), of a graph G is the intersection graph of the maximal cliques of G. For a natural number n, a graph G is n-convergent if Kn(G) is isomorphic to K1 (the one-vertex graph). A graph G is convergent if it is n-convergent for some natural number n. A 2-convergent graph is called clique-complete. We describe the family of minimal graphs which are clique-complete but have no universal vertices. The minimality used here refers to induced subgraphs. In addition, we show that recognizing clique-complete graphs is Co-NP-complete.