{
  "id": "2307.08447",
  "version": "v1",
  "published": "2023-07-17T12:42:21.000Z",
  "updated": "2023-07-17T12:42:21.000Z",
  "title": "The skeleton of a convex polytope",
  "authors": [
    "Takayuki Hibi",
    "Aki Mori"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let ${\\rm sk}({\\mathcal P})$ denote the $1$-skeleton of an convex polytope ${\\mathcal P}$. Let $C$ be a clique (=complete subgraph) of ${\\rm sk}({\\mathcal P})$ and ${\\rm conv}(C)$ the convex hull of the vertices of ${\\mathcal P}$ belonging to $C$. In general, ${\\rm conv}(C)$ may not be a face of ${\\mathcal P}$. It will be proved that ${\\rm conv}(C)$ is a face of ${\\mathcal P}$ if ${\\mathcal P}$ is either the order polytope ${\\mathcal O}(P)$ of a finite partially ordered set $P$ or the stable set polytope ${\\rm Stab}(G)$ of a finite simple graph $G$. In other words, when ${\\mathcal P}$ is either ${\\mathcal O}(P)$ or ${\\rm Stab}(G)$, the simplicial complex consisting of simplices which are faces of ${\\mathcal P}$ is the clique complex of ${\\rm sk}({\\mathcal P})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-17T12:42:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11"
    ],
    "keywords": [
      "convex polytope",
      "finite simple graph",
      "simplicial complex",
      "convex hull",
      "clique complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}