The clique operator K maps a graph G into its clique graph, which is the intersection graph of the (maximal) cliques of G. Among all the better studied graph operators, K seems to be the richest one and many questions regarding it remain open. In particular, it is not known whether recognizing a clique graph is in P. In this note we describe our progress toward answering this question. We obtain a necessary condition for a graph to be in the image of K in terms of the presence of certain subgraphs A and B. We show that being a clique graph is not a property that is maintained by addition of twins. We present a result involving distances that reduces the recognition problem to graphs of diameter at most two. We also give a constructive characterization of K−1(G) for a fixed but generic G.