Complexes of directed trees and independence complexes

First we prove that certain complexes on directed acyclic graphs are shellable. Then we study independence complexes. Two theorems used for breaking and gluing such complexes are proved and applied to generalize the results by Kozlov.An interesting special case is anti-Rips complexes: a subset PP of a metric space is the vertex set of the complex, and we include as a simplex each subset of PP with no pair of points within distance rr. For any finite subset PP of RR the homotopy type of the anti-Rips complex is determined.