{
  "id": "math/0202103",
  "version": "v1",
  "published": "2002-02-12T09:46:40.000Z",
  "updated": "2002-02-12T09:46:40.000Z",
  "title": "Reconstructing a Simple Polytope from its Graph",
  "authors": [
    "Volker Kaibel"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "math.MG"
  ],
  "abstract": "Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling order of the facets of the dual polytope of P). Thereby, we derive polynomial certificates for both the vertex-facet incidences as well as for the abstract objective functions in terms of the graph of P. The paper is a variation on joint work with Michael Joswig and Friederike Koerner (2001).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-02-12T09:46:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11",
      "52B05",
      "52B22"
    ],
    "keywords": [
      "simple polytope",
      "vertex-facet incidences",
      "abstract objective function",
      "reconstruct",
      "entire combinatorial structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2103K"
    }
  }
}