A Discrete Morse Theory for Digraphs

Chong Wang; Shiquan Ren

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arXiv · 2007.13425

A Discrete Morse Theory for Digraphs

Chong Wang·Shiquan Ren

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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.

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Chong Wang, Shiquan Ren. 2020-07-27. A Discrete Morse Theory for Digraphs. https://arxiv.org/abs/2007.13425

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