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A multipath in a directed graph is a disjoint union of paths. The multipathcomplex of a directed graph ${\tt G}$ is the simplicial complex whose faces arethe multipaths of ${\tt G}$. We compute the Euler characteristic, andassociated generating function, of the multipath complex for some families ofgraphs, including transitive tournaments and complete bipartite graphs. Then,we compute the homotopy type of multipath complexes of linear graphs, polygons,small grids and transitive tournaments. We show that they are all contractibleor wedges of spheres. We introduce a new technique for decomposing directedgraphs into dynamical regions, which allows us to simplify the homotopycomputations.

On the homotopy type of multipath complexes