Simplicial simple-homotopy of flag complexes in terms of graphs

Abstract

A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian ([2]) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the s-homotopy and the graph homotopy defined in [5].

Clique complexes and graph powers

“…A graph G is dismantlable if there exists a sequence of folds from G to a single vertex. It is a classical fact that a fold preserves the homotopy type of the clique complex and, in fact, induces a collapsing of Cl(G) onto Cl(G \ u), so the clique complex of a dismantlable graph is collapsible (see for example [4,Lemma 2.2]). In this context we have the following simple result.…”

“…We will identify the vertices of T n,k with Z/n. Under this identification each vertex i is connected to the vertices in the set (4) N T n,k (i) = {i + r + 1, . .…”

Four-Cycled Graphs with Topological Applications

“…In a recent paper, Boulet, Fieux, and Jouve [3] have introduced the notion of shomotopy type of a graph through an operation called s-dismantling. Following Barmak and Minian's work on posets [2], they completely characterize combinatorially how to detect the simple-homotopy type of the clique complex of a given graph.…”

“…These ideas are followed by Boulet, Fieux, and Jouve [3], where they introduce the notion of s-homotopy type of graphs through an operation called s-dismantling. They derive an analogous result to that of Barmak and Minian regarding the simplehomotopy types of clique complexes, namely, two clique complexes are simplehomotopic if and only if the corresponding graphs are s-homotopic.…”

Algebraic Topology of Finite Topological Spaces and Applications

“…Applications of our methods and results to graph theory can be found in [19]. A relationship of finite spaces with toric varieties is discussed in [12].…”