On clique-inverse graphs of graphs with bounded clique number

The clique graph K(G) of G is the intersection graph of the family of maximal cliques of G. For a family F of graphs, the family of clique-inverse graphs of F, denoted by K-1(F), is defined as K-1(F)={H|K(H)is an element of F}. Let Fp be the family of K-p-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 Fp can be done in polynomial time; in addition, for p is an element of{2,3,4},K-1(Fp) 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 K-1(Fp) 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 K-1(Fp) in terms of p.