The complexity of clique graph recognition

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C (G) the clique family of G. The clique graph of G, denoted by K (G), is the intersection graph of C (G). Say that G is a clique graph if there exists a graph H such that G = K (H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. However, the time complexity of the problem of recognizing clique graphs is a long-standing open question. We prove that the clique graph recognition problem is NP-complete.Facultad de Ciencias Exacta

The complexity of clique graph recognition

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C (G) the clique family of G. The clique graph of G, denoted by K (G), is the intersection graph of C (G). Say that G is a clique graph if there exists a graph H such that G = K (H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. However, the time complexity of the problem of recognizing clique graphs is a long-standing open question. We prove that the clique graph recognition problem is NP-complete.Facultad de Ciencias Exacta

The complexity of clique graph recognition

A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set. Denote by C (G) the clique family of G. The clique graph of G, denoted by K (G), is the intersection graph of C (G). Say that G is a clique graph if there exists a graph H such that G = K (H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. However, the time complexity of the problem of recognizing clique graphs is a long-standing open question. We prove that the clique graph recognition problem is NP-complete.Facultad de Ciencias Exacta

The Clique Operator On Graphs With Few P4's

The clique graph of a graph G is the intersection graph K(G) of the (maximal) cliques of G. The iterated clique graphs Kn(G) are defined by K0(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 P4's . These characterizations lead to polynomial time algorithms for deciding the K-convergence or K-divergence of any graph in the class. © 2005 Elsevier B.V. All rights reserved.1543485492Babel, L., Kloks, T., Kratochvíl, J., Kratsch, D., Müller, H., Olariu, S., Efficient algorithms for graphs with few P4's (2001) Discrete Math., 235, pp. 29-51Bandelt, H.-J., Prisner, E., Clique graphs and Helly graphs (1991) J. Combin. Theory Ser. B, 51, pp. 34-45Bornstein, C.F., Szwarcfiter, J.L., On clique convergent graphs (1995) Graphs Combin., 11, pp. 213-220Cheen, B.-L., Lih, K.-W., Diameters of iterated clique graphs of chordal graphs (1990) J. Graph Theory, 14, pp. 391-396Cournier, A., Habib, M., A new linear algorithms for modular decomposition (1994) Lecture Notes in Computer Science, 787, pp. 68-84. , Springer, BerlinEscalante, F., Über iterierte clique-graphen (1973) Abh. Math. Sem. Univ. Hamburg, 39, pp. 59-68Frias, M.E., Neumann-Lara, V., Pizaña, M., Dismantlings and iterated clique graphs (2004) Discrete Math., 282, pp. 263-265Giakoumakis, V., Vanherpe, J.-M., On extended P4P-reducible and extended P4-sparse graphs (1997) Theoret. Comput. Sci., 180, pp. 269-286Giakoumakis, V., Roussel, F., Thuillier, H., On P4-tidy graphs (1997) Discrete Math. Theoret. Comput. Sci., 1, pp. 17-41Hamelink, R., A partial characterization of clique graphs (1968) J. Combin. Theory, 5, pp. 192-197Hoang, C., (1985), Doctoral Dissertation, McGill University Montreal, QuebecJamison, B., Olariu, S., A new class of brittle graphs (1989) Stud. Appl. Math., 81, pp. 89-92Jamison, B., Olariu, S., P4-reducible graphs, a class of uniquely tree representable graphs (1989) Stud. Appl. Math., 81, pp. 79-87Jamison, B., Olariu, S., On a unique tree representation for P4-extendible graphs (1991) Discrete Appl. Math., 34, pp. 151-164Jamison, B., Olariu, S., A tree representation for P4-sparse graphs (1992) Discrete Appl. Math., 35, pp. 115-129Larrión, F., De Mello, C.P., Morgana, A., Neumann-Lara, V., Pizaña, M., The clique operator on cographs and serial graphs (2004) Discrete Math., 282, pp. 183-191Larrión, F., Neumann-Lara, V., Clique divergent graphs with unbounded sequence of diameter (1999) Discrete Math., 197-198, pp. 491-501Larrión, F., Neumann-Lara, V., Locally C6 graphs are clique divergent (2000) Discrete Math., 215, pp. 159-170Larrión, F., Neumann-Lara, V., On clique divergent graphs with linear growth (2001) Discrete Math., 245, pp. 139-153Larrión, F., Neumann-Lara, V., Pizaña, M., Whitney triangulations, local girth and iterated clique graphs (2002) Discrete Math., 258, pp. 123-135McConnell, R.M., Spinrad, J.P., Modular decomposition and transitive orientation (1999) Discrete Math., 201, pp. 189-206Neumann-Lara, V., On clique-divergent graphs (1978) Colloq. Internat. CNRS, 260, pp. 313-315. , Problèmes Combinatoires et Théorie des GraphesNeumann-Lara, V., Clique-divergence in graphs (1978) Coll. Math. Soc. Janos Bolyai, 25, pp. 563-569. , Algebraic Methods in Graph Theory, Szeged, Húngary North-Holland, AmsterdamNeumann-Lara, V., Theory of Clique Expansive Graphs, , unpublished manuscriptPizaña, M.A., The icosahedron is clique divergent (2003) Discrete Math., 262, pp. 229-239Prisner, E., Convergence of iterated clique graphs (1992) Discrete Math., 103, pp. 199-207Prisner, E., Hereditary Helly graphs (1993) J. Combin. Math. Combin. Comput., 14, pp. 216-220Prisner, E., Graph dynamics (1995) Pitman Research Notes in Mathematics, 338. , Longman, New YorkSzwarcfiter, J.L., Recognizing clique-Helly graphs (1997) Ars. Combin., 45, pp. 29-32Szwarcfiter, J.L., A survey on clique graphs (2003) Recent Advances in Algorithms and Combinatorics, CMS Books in Mathematics, pp. 109-136. , C. Linhares B. Reed Springer Berli