{
  "id": "math/0011170",
  "version": "v2",
  "published": "2000-11-22T00:22:06.000Z",
  "updated": "2001-07-19T11:50:05.000Z",
  "title": "Examples and counterexamples for Perles' conjecture",
  "authors": [
    "Christian Haase",
    "Günter M. Ziegler"
  ],
  "comment": "11 pages, 14 figures, see also http://www.math.tu-berlin.de/~ziegler",
  "categories": [
    "math.CO"
  ],
  "abstract": "The combinatorial structure of a d-dimensional simple convex polytope can be reconstructed from its abstract graph [Blind & Mani 1987, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction was found by [Joswig, Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of the graph of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we identify a topological obstruction that must be present in any counterexample to Perles' conjecture; thus, starting with a modification of ``Bing's house'', we construct explicit 4-dimensional counterexamples.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-07-19T11:50:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B05",
      "05C75"
    ],
    "keywords": [
      "counterexample",
      "d-dimensional simple convex polytope",
      "facet subgraphs",
      "construct explicit",
      "abstract graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math.....11170H"
    }
  }
}