A Discrete Morse Theory for Digraphs

TL;DR

This paper introduces a discrete Morse theory framework for digraphs, establishing homology isomorphisms between Morse complexes and path homology, and analyzing collapses via discrete gradient vector fields.

Contribution

It defines discrete Morse functions on digraphs and proves homology isomorphisms for transitive digraphs and certain collapses, extending Morse theory to directed graphs.

Findings

Homology of Morse complex is isomorphic to path homology for transitive digraphs.

Original digraph and its collapse share the same path homology groups.

Discrete gradient vector fields induce collapses preserving homology.

Abstract

Digraphs are generalizations of graphs in which each edge is assigned with a direction or two directions. In this paper, we define discrete Morse functions on digraphs, and prove that the homology of the Morse complex and the path homology are isomorphic for a transitive digraph. We also study the collapses defined by discrete gradient vector fields. Let $G$ be a digraph and $f$ a discrete Morse function. Assume the out-degree and in-degree of any zero-point of $f$ on $G$ are both 1. We prove that the original digraph $G$ and its $\mathcal{M}$-collapse $\tilde{G}$ have the same path homology groups.