Simplicial simple-homotopy of flag complexes in terms of graphs

TL;DR

This paper introduces s-dismantlability to define s-homotopy types of graphs and demonstrates their equivalence to the simple-homotopy types of associated flag complexes, linking graph theory and topological concepts.

Contribution

It establishes a new framework connecting graph s-homotopy with simplicial simple-homotopy of flag complexes, extending previous poset-based results.

Findings

s-homotopy type of a graph matches the simple-homotopy type of its flag complex

relation between s-homotopy and poset-based approaches

raises questions about connection to graph homotopy by Chen, Yau, and Yeh

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 (Adv. in Math., 218 (2008), 87-104) 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 by Chen, Yau and Yeh (Discrete Math., 241(2001), 153-170).