Miguel A. Pizaña

Whitney triangulations, local girth and iterated clique graphs

We study the dynamical behaviour of surface triangulations under the iterated application of the clique graph operator k, which transforms each graph G into the intersection graph kG of its (maximal) cliques. A graph G is said to be k-divergent if the sequence of the orders of its iterated clique graphs |V (k n G)| tends to in4nity with n. If this is not the case, then G is eventually k-periodic, or k-bounded: k n G ∼ k m G for some m?n . The case in which G is the underlying graph of a regular triangulation of some closed surface has been previously studied under the additional (Whitney) hypothesis that every triangle of G is a face of the triangulation: if G is regular of degree d, it is known that G is k-bounded for d = 3 and k-divergent for d = 4; 5; 6. We will show that G is k-bounded for all d ? 7, thus completing the study of the regular case. Our proof works in the more general setting of graphs with local girth at least 7. As a consequence we obtain also the k-boundedness of the underlying graph G of any triangulation of a compact surface (with or without border) provided that any triangle of G is a face of the triangulation and that the minimum degree of the interior vertices of G is at least 7. c � 2002 Published by Elsevier Science B.V.

The icosahedron is clique divergent

A 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 Larrion and Neumann-Lara in 2000. This paper proves the clique divergence of the icosahedron among other results of general interest in clique divergence theory.

Clique divergent clockwork graphs and partial orders

S. Hazan and V. Neumann-Lara proved in 1996 that every finite partially ordered set whose comparability graph is clique null has the fixed point property and they asked whether there is a finite poset with the fixed point property whose comparability graph is clique divergent. In this work we answer that question by exhibiting such a finite poset. This is achieved by developing further the theory of clockwork graphs. We also show that there are polynomial time algorithms that recognize clockwork graphs and decide whether they are clique divergent.

On expansive graphs

The clique graph K(G) of a graph G, is the intersection graph of its (maximal) cliques, and G is K-divergent if the orders of its iterated clique graphs K(G),K^2(G),K^3(G),... tend to infinity. A coaffine graph has a symmetry that maps each vertex outside of its closed neighbourhood. For these graphs we study the notion of expansivity, which implies K-divergence.

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K^0(G)=G, K^n^+^1(G)=K(K^n(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that K^n(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G~K^n(G) for all n, moreover K^n(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.

The fundamental group of the clique graph

Given a finite connected bipartite graph B=(X,Y) we consider the simplicial complexes of complete subgraphs of the square B^2 of B and of its induced subgraphs B^2[X] and B^2[Y]. We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K(G), the line graph L(G) and the total graph T(G).

Contractibility and the clique graph operator

To any graph G we can associate a simplicial complex @D(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever @D(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whose clique graph is a cone is contractible.

Posets, clique graphs and their homotopy type

To any finite poset P we associate two graphs which we denote by @W(P) and @?(P). Several standard constructions can be seen as @W(P) or @?(P) for suitable posets P, including the comparability graph of a poset, the clique graph of a graph and the 1-skeleton of a simplicial complex. We interpret graphs and posets as simplicial complexes using complete subgraphs and chains as simplices. Then we study and compare the homotopy types of @W(P), @?(P) and P. As our main application we obtain a theorem, stronger than those previously known, giving sufficient conditions for a graph to be homotopy equivalent to its clique graph. We also introduce a new graph operator H that preserves clique-Hellyness and dismantlability and is such that H(G) is homotopy equivalent to both its clique graph and the graph G.

Distances and diameters on iterated clique graphs

If G is a graph, its clique graph, K(G), is the intersection graph of all its (maximal) cliques. Iterated clique graphs are then defined recursively by: K0(G)=G and Kn(G)=K(Kn-1(G)). We study the relationship between distances in G and distances in Kn(G). Then we apply these results to Johnson graphs to give a shorter and simpler proof of Bornstein and Szwarefiters theorem: For each n there exists a graph G such that diam(Kn(G))=diam(G)+n. In the way, a new family of graphs with increasing diameters under the clique operator is shown.

On hereditary clique-Helly self-clique graphs

A graph is clique-Helly if any family of mutually intersecting (maximal) cliques has non-empty intersection, and it is hereditary clique-Helly (HCH) if its induced subgraphs are clique-Helly. The clique graph of a graph G is the intersection graph of its cliques, and G is self-clique if it is connected and isomorphic to its clique graph. We show that every HCH graph is an induced subgraph of a self-clique HCH graph, and give a characterization of self-clique HCH graphs in terms of their constructibility starting from certain digraphs with some forbidden subdigraphs. We also specialize this results to involutive HCH graphs, i.e. self-clique HCH graphs whose vertex-clique bipartite graph admits a part-switching involution.