On the homotopy type of multipath complexes

A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph $G$ is the simplicial complex whose faces are the multipaths of $G$. We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show that if $G$ is a linear graph, polygon, small grid or transitive tournament, then the homotopy type of the multipath complex of $G$ is always contractible or a wedge of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.