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 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 R the homotopy type of the anti-Rips complex is determined. © 2008 Elsevier B.V. All rights reserved.
Engström, A. (2009). Complexes of directed trees and independence complexes. Discrete Mathematics, 309(10), 3299–3309. https://doi.org/10.1016/j.disc.2008.09.033