Shellability of complexes of directed trees

Abstract

The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the crosspolytopes allow such a shelling. Furthermore, we show that the complex of directed trees of a complete double directed graph is a union of suitable spheres. We prove that the complex of directed trees of a directed graph which is essentially a tree is vertex-decomposable. For these complexes we describe their sets of generating facets.