On the homotopy type of the clique graph

Abstract

­ 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?]]>

Posets, clique graphs and their homotopy type

“…For instance, under a mild condition on P , Ω(P ) has the same homotopy type as the clique graph of (P ), see Theorem 3.3. Furthermore, Theorem 5.7 generalizes the main result in [8], which was the strongest result asserting the homotopy equivalence of a graph and its clique graph. As we shall see, an interesting feature of Ω and is that, combined with standard constructions on posets, graphs and simplicial complexes, they yield several well known constructions, thus providing a unified approach to them.…”

“…2.4] is equivalent to P b (P(G)) ≤h(C) being conically contractible for all completes C of K(G). As in [8], Theorem 5.7 implies that the only Whitney triangulation of a closed surface which is not homotopy equivalent to its clique graph is the octahedron. Here, a Whitney triangulation of a surface S is a graph G such that |∆(G)| ∼ = S.…”

“…It is known that G and K(G) are not always homotopy equivalent [10]. The motivations for this work came from two fronts: Poset topology, as in [11,13,3], and homotopy type of clique graphs, as in [10,8]. In this work, we are interested in comparing the homotopy types of P and Ω(P ), (P ) and their clique graphs.…”

The clique operator on matching and chessboard graphs

“…Larrión, Neumann-Lara and Pizaña [17] gave a condition that shows that many graphs G which are not clique-Helly still satisfy 17]). Let G be such that each complete X of K (G) with ∩X = ∅ has a center that is contained in every necktie that contains X .…”

“…However, many non-clique-Helly graphs G still satisfy K (G) G. We show in Section 4 that some non-clique-Helly matching and chessboard graphs have this property. In order to do this we shall use a generalization of Prisner's result due to Larrión, Neumann-Lara and Pizaña [17] and a further similar result (see 4.2) of our own.…”