A Simple Uniform Approach to Complexes Arising from Forests

In the recent years several complexes arising from forests have been studied by different authors with different techniques (see [BM], [BLN], [EH], [E], [Kl], [K1], [K2], [MT], [W]). The interest in these problems is motivated by applications in different contexts, such as graph theory and statistical mechanics ([BK], [BLN], [J]). We introduce a unifying approach to study the homotopy type of several of these complexes. With our technique we obtain simple proofs of results that were already known as well as new results. These complexes are wedges of spheres of (possibly) different dimensions and include, for instance, the complexes of directed trees, the independence complexes, the dominance complexes, the matching complexes, the interval order complexes. In all cases our method provides a recursive procedure to compute the exact homotopy type of the simplicial complex. The dimensions of the spheres arising with these constructions are often strictly related to classical graph theoretical invariants of the underlying forest. Thus we give a topological interpretation to these well-known combinatorial invariants. Section 2 is devoted to notation and background. In Section 3 we introduce the two basic concepts of this paper: the simplicial complex property of being a grape and the strictly related notion of domination between vertices of a simplicial complex. In Section 4 we discuss several applications of these notions.