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.
Given positive integers m, n, we consider the graphs G n and G m,n whose simplicial complexes of complete subgraphs are the well-known matching complex M n and chessboard complex M m,n . Those are the matching and chessboard graphs. We determine which matching and chessboard graphs are clique-Helly. If the parameters are small enough, we show that these graphs (even if not clique-Helly) are homotopy equivalent to their clique graphs. We determine the clique behavior of the chessboard graph G m,n in terms of m and n, and show that G m,n is clique-divergent if and only if it is not clique-Helly. We give partial results for the clique behavior of the matching graph G n .
To any finite poset P we associate two graphs which we denote by Ω(P) and (P). Several standard constructions can be seen as Ω(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 Ω(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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.