Simplicial simple-homotopy of flag complexes in terms of graphs
arXiv:0809.1751
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).
15 pages, 8 figures (in tex format; uses pstricks). In version v2, the hypothesis of the Lemma 4.4 was incomplete. This has been corrected in version v3 and, as a consequence, Proposition 4.5 states only an implication and not an equivalence. These corrections are the subject of the erratum https://doi.org/10.1016/j.ejc.2019.05.004 in European Journal of Combinatorics