Complexes of Directed Trees

To every directed graph G one can associate a complex Δ(G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn, where it leads to the study of some interesting representations of the symmetric group and corresponds (via the Stanley–Reisner correspondence) to an interesting quotient ring. Our main result states that Δ(Gn) is shellable, in particular, Cohen–Macaulay, which can be further translated to say that the Stanley–Reisner ring of Δ(Gn) is Cohen–Macaulay. Besides that, by computing the homology groups of Δ(G) for the cases when G is essentially a tree and when G is a double directed cycle, we touch upon the general question of the interaction of the combinatorial properties of a graph and the topological properties of the associated complex.