{
  "id": "2011.12225",
  "version": "v1",
  "published": "2020-11-24T17:24:35.000Z",
  "updated": "2020-11-24T17:24:35.000Z",
  "title": "Completing and extending shellings of vertex decomposable complexes",
  "authors": [
    "Michaela Coleman",
    "Anton Dochtermann",
    "Nathan Geist",
    "Suho Oh"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "We say that a pure $d$-dimensional simplicial complex $\\Delta$ on $n$ vertices is shelling completable if $\\Delta$ can be realized as the initial sequence of some shelling of $\\Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex. A well-known conjecture of Simon posits that any shellable complex is shelling completable. In this note we prove that vertex decomposable complexes are shelling completable. In fact we show that if $\\Delta$ is a vertex decomposable complex then there exists an ordering of its ground set $V$ such that adding the revlex smallest missing $(d+1)$-subset of $V$ results in a complex that is again vertex decomposable. We explore applications to matroids, shifted complexes, as well as $k$-vertex decomposable complexes. We also show that if $\\Delta$ is a $d$-dimensional complex on at most $d+3$ vertices then the notions of shellable, vertex decomposable, shelling completable, and extendably shellable are all equivalent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-24T17:24:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "52B22",
      "52B40",
      "13F55"
    ],
    "keywords": [
      "vertex decomposable complex",
      "extending shellings",
      "shelling completable",
      "dimensional simplicial complex",
      "initial sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}