The fundamental group of the clique graph

AbstractGiven a finite connected bipartite graph B=(X,Y) we consider the simplicial complexes of complete subgraphs of the square B2 of B and of its induced subgraphs B2[X] and B2[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)

Posets, Clique Graphs and their Homotopy Type

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. Key words: clique graphs, graphs, posets, homotopy type

ON HEREDITARY CLIQUE HELLY SELF-CLIQUE GRAPHS

NOTICE: this is the author’s version of a work that was accepted for publication in Discrete Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication

THE CLIQUE OPERATOR ON MATCHING AND CHESSBOARD GRAPHS

Abstract. Given positive integers m, n, we consider the graphs Gn and Gm,n whose simplicial complexes of complete subgraphs are the well-known matching complex Mn and chessboard complex Mm,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 Gm,n in terms of m and n, and show that Gm,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 Gn. 1

CONTRACTIBILITY AND THE CLIQUE GRAPH OPERATOR

AbstractTo any graph G we can associate a simplicial complex Δ(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever Δ(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