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Given a graph G and two vertices x; y ∈ V (G), we say that x is dominated by y if the closed neighbourhood of x is contained in that of y. Here we prove that if x is a dominated vertex, then G and G − {x} have the same dynamical behaviour under the iteration of the clique operator.

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Abstract. This work has two aims: First, we introduce a powerful technique for proving clique divergence when the graph satisfies a certain symmetry condition. Second, we prove that each closed surface admits a clique divergent triangulation. By definition, a graph is clique divergent if the orders of its iterated clique graphs tend to infinity, and the clique graph of a graph is the intersection graph of its maximal complete subgraphs.

­ of a graph G is the simplicial complex whose simplexes are the vertex sets of the complete subgraphs of G. Here we study a sufficient condition for G and K(G) to be homotopic. Applying this result to Whitney triangulations of surfaces, we construct an infinite family of examples which solve in the affirmative Prisner's open problem 1 in Graph Dynamics (Longman, Harlow, 1995): Are there finite connected graphs G that are periodic under K and where the second modulo 2 Betti number is greater than 0?]]>

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