Directed Subgraph Complexes

Let G be a directed graph, and let ∆ACY G be the simplicial complex whose simplices are the edge sets of acyclic subgraphs of G. Similarly, we define ∆NSC G to be the simplicial complex with the edge sets of not strongly connected subgraphs of G as simplices. We show that ∆ACY G is homotopy equivalent to the (n−1−k)-dimensional sphere if G is a disjoint union of k strongly connected graphs. Otherwise, it is contractible. If G belongs to a certain class of graphs, the homotopy type of ∆NSC G is shown to be a wedge of (2n − 4)-dimensional spheres. The number of spheres can easily be read off the chromatic polynomial of a certain associated undirected graph. We also consider some consequences related to finite topologies and hyperplane arrangements.