The clique operator on graphs with few P4P4's

The clique graph of a graph G   is the intersection graph K(G)K(G) of the (maximal) cliques of G  . The iterated clique graphs Kn(G)Kn(G) are defined by K0(G)=GK0(G)=G and Ki(G)=K(Ki-1(G)),i>0 and K is the clique operator. In this article we use the modular decomposition technique to characterize the K  -behaviour of some classes of graphs with few P4P4's . These characterizations lead to polynomial time algorithms for deciding the K-convergence or K-divergence of any graph in the class.