On the homotopy type of multipath complexes

TL;DR

This paper investigates the topological properties of multipath complexes in directed graphs, computing their homotopy types and Euler characteristics for various graph families, and introduces a new decomposition technique.

Contribution

It provides the first detailed homotopy type analysis of multipath complexes for several graph classes and introduces a novel graph decomposition method.

Findings

Multipath complexes of linear graphs, polygons, small grids, and transitive tournaments are contractible or wedges of spheres.

Euler characteristics and generating functions for multipath complexes are explicitly computed for certain graph families.

A new technique for decomposing directed graphs into dynamical regions simplifies homotopy computations.

Abstract

A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph ${\tt G}$ is the simplicial complex whose faces are the multipaths of ${\tt G}$. We compute the Euler characteristic, and associated generating function, of the multipath complex for some families of graphs, 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 contractible or wedges of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.