The clique graph of a graph G is the intersection graph K(G) of the (maximal) cliques of G. The iterated clique graphs K(n)(G) are defined by K(0)(G) = G and K(i)(G) = K(K(i-1)(G)), i > 0 and K is the clique operator. A cograph is a graph with no induced subgraph isomorphic to P(4). In this article we use the modular decomposition technique to characterize the K-behaviour of cographs and to give some partial results for the larger class of serial (i.e. complement-disconnected) graphs. We prove that a cograph is K-convergent if and only if it is clique-Helly. This characterization leads to a polynomial time algorithm for deciding the K-convergence or K-divergence of any cograph. (C) 2003 Elsevier B.V. All rights reserved.28241699183191