{
  "id": "1811.08031",
  "version": "v1",
  "published": "2018-11-20T00:26:12.000Z",
  "updated": "2018-11-20T00:26:12.000Z",
  "title": "Combinatorial modifications of Reeb graphs and the realization problem",
  "authors": [
    "Łukasz Patryk Michalak"
  ],
  "comment": "21 pages",
  "journal": "Discrete Comput. Geom. 65 (2021), 1038-1060",
  "doi": "10.1007/s00454-020-00260-6",
  "categories": [
    "math.GT"
  ],
  "abstract": "We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the number of cycles can be realized as the Reeb graph of a Morse function on a given closed manifold $M$. Along the way, we show that the Reeb number $\\mathcal{R}(M)$, i.e. the maximal number of cycles among all Reeb graphs of functions on $M$, is equal to the corank of fundamental group $\\pi_1(M)$, thus extending a previous result of Gelbukh to the non-orientable case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-20T00:26:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C76",
      "57M15",
      "05C38"
    ],
    "keywords": [
      "reeb graph",
      "combinatorial modifications",
      "realization problem",
      "natural necessary conditions",
      "reeb number"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}