The icosahedron is clique divergent

AbstractA clique of a graph G is a maximal complete subgraph. The clique graph k(G) is the intersection graph of the set of all cliques of G. The iterated clique graphs are defined recursively by k0(G)=G and kn+1(G)=k(kn(G)). A graph G is said to be clique divergent (or k-divergent) if limn→∞|V(kn(G))|=∞. The problem of deciding whether the icosahedron is clique divergent or not was (implicitly) stated Neumann-Lara in 1981 and then cited by Neumann-Lara in 1991 and Larrión and Neumann-Lara in 2000. This paper proves the clique divergence of the icosahedron among other results of general interest in clique divergence theory

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