Complexes of Directed Trees and Independence Complexes

The theory of complexes of directed trees was initiated by Kozlov to answer a question by Stanley, and later on, results from the theory were used by Babson and Kozlov in their proof of the Lovasz conjecture. We develop the theory and prove that complexes on directed acyclic graphs are shellable. A related concept is that of independence complexes: construct a simplicial complex on the vertex set of a graph, by including each independent set of vertices as a simplex. Two theorems used for breaking and gluing such complexes are proved and applied to generalize results by Kozlov. A fruitful restriction is anti-Rips complexes: a subset P of a metric space is the vertex set of the complex, and we include as a simplex each subset of P with no pair of points within distance r. For any finite subset P of \mathbb{R} the homotopy type of the anti-Rips complex is determined.Comment: 14 pages, 4 figure

A simple uniform approach to complexes arising from forests

In this paper we present a unifying approach to study the homotopy type of several complexes arising from forests. We show that this method applies uniformly to many complexes that have been extensively studied.Comment: 13 pages. Comments very welcom

Shellability of complexes of directed trees

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 also investigate shellability of the maximal pure skeleton of a complex of directed trees. Also, we prove that is vertex-decomposable. For these complexes we describe the set of generating facets.Comment: This is a new version in which Section 3 about complexes $\mathcal{C}_n^k$ is removed. There are some troubles in the proof of Theorem 14. We add a new section about the complexes of directed trees of a directed graph which is essentially a tre