Complexes of Directed Graphs 3 2

Let P be a monotone property of directed graphs on n vertices, and let P n denote the abstract simplicial complex whose simplices are the edge sets of graphs having property P. We prove that: 1. If \P = acyclic" then P n is homotopy equivalent to the (n ? 2)-sphere. 2. If \P = not strongly connected" then P n has the homotopy type of a wedge of (n ? 1)! spheres of dimension 2n ? 4. The lattice of all posets on f1;2;: : : ; ng plays an important role in the analysis. We also discuss a few other properties of directed graphs from this point of view. 1. Introduction A property of graphs is called monotone if it is preserved under deletion of edges. Thus, a monotone graph property can be interpreted as a simplicial complex, and one can study its topological properties. This has been done for undirected graphs in a number of recent papers. See 1, 13] and the further references cited there. Here we look at some monotone properties of directed graphs from this point of