{
  "id": "2007.13425",
  "version": "v1",
  "published": "2020-07-27T11:01:28.000Z",
  "updated": "2020-07-27T11:01:28.000Z",
  "title": "A Discrete Morse Theory for Digraphs",
  "authors": [
    "Chong Wang",
    "Shiquan Ren"
  ],
  "categories": [
    "math.AT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-27T11:01:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55U15",
      "55N35",
      "55U35"
    ],
    "keywords": [
      "discrete morse theory",
      "discrete gradient vector fields",
      "define discrete morse functions",
      "path homology groups",
      "morse complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}