Clique-critical graphs: Maximum size and recognition

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K−1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170–182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete. © 2006 Elsevier B.V. All rights reserved.