{
  "id": "2001.06775",
  "version": "v1",
  "published": "2020-01-19T05:01:00.000Z",
  "updated": "2020-01-19T05:01:00.000Z",
  "title": "Distance $r$-domination number and $r$-independence complexes of graphs",
  "authors": [
    "Priyavrat Deshpande",
    "Samir Shukla",
    "Anurag Singh"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "For $r\\geq 1$, the $r$-independence complex of a graph $G$, denoted Ind$_r(G)$, is a simplicial complex whose faces are subsets $A \\subseteq V(G)$ such that each component of the induced subgraph $G[A]$ has at most $r$ vertices. In this article, we establish a relation between the distance $r$-domination number of $G$ and (homological) connectivity of Ind$_r(G)$. We also prove that Ind$_r(G)$, for a chordal graph $G$, is either contractible or homotopy equivalent to a wedge of spheres. Given a wedge of spheres, we also provide a construction of a chordal graph whose $r$-independence complex has the homotopy type of the given wedge.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-19T05:01:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "55P15"
    ],
    "keywords": [
      "independence complex",
      "domination number",
      "chordal graph",
      "homotopy type",
      "homotopy equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}