On clique‐inverse graphs of graphs with bounded clique number

Abstract The clique graph K(G) of G is the intersection graph of the family of maximal cliques of G. For a family of graphs, the family of clique‐inverse graphs of , denoted by , is defined as . Let be the family of Kp‐free graphs, that is, graphs with clique number at most p − 1, for an integer constant p ≥ 2. Deciding whether a graph H is a clique‐inverse graph of can be done in polynomial time; in addition, for can be characterized by a finite family of forbidden induced subgraphs. In Protti and Szwarcfiter, the authors propose to extend such characterizations to higher values of p. Then a natural question arises: Is there a characterization of by means of a finite family of forbidden induced subgraphs, for any p ≥ 2? In this note we give a positive answer to this question. We present upper bounds for the order, the clique number, and the stability number of every forbidden induced subgraph for in terms of p.