s-homotopy for finite graphs

We introduce the notion of “s-dismantlability” which will give in the category of finite graphs an analogue of formal deformations defining the simple-homotopy type in the category of finite simplicial complexes. More precisely, s-dismantlability allows us to define an equivalence relation whose equivalence classes are called “s-homotopy types” and we get a correspondence between s-homotopy types in the category of graphs and simple-homotopy types in the category of simplicial complexes by the way of classical functors between these two categories (theorem 3.6). Next, we relate these results to similar results obtained by Barmak and Minian (2006) within the framework of posets (theorem 4.2).