Simplicial simple-homotopy of flag complexes in terms of graphs

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. This result is closely related to similar results established by Barmak and Minian [J.A. Barmak, E.G. Minian, Simple homotopy types and finite spaces, Adv. Math. 218 (1) (2008) 87–104. doi:10.1016/j.aim.2007.11.019] in the framework of posets and we give the relation between the two approaches. We conclude with a question about the relation between the s-homotopy and the graph homotopy defined in [B. Chen, S.-T. Yau, Y.-N. Yeh, Graph homotopy and Graham homotopy, Selected papers in honor of Helge Tverberg, Discrete Math. 241 (1-3) (2001) 153–170. doi:10.1016/S0012-365X(01)00115-7.]