Equivariant collapses and the homotopy type of iterated clique graphs

Abstract

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.