{
  "id": "2109.04181",
  "version": "v1",
  "published": "2021-09-09T11:37:34.000Z",
  "updated": "2021-09-09T11:37:34.000Z",
  "title": "Independence Complex of the Lexicographic Product of a Forest",
  "authors": [
    "Kengo Okura"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the independence complex of the lexicographic product $G[H]$ of a forest $G$ and a graph $H$. We prove that for a forest $G$ which is not dominated by a single vertex, if the independence complex of $H$ is homotopy equivalent to a wedge sum of spheres, then so is the independence complex of $G[H]$. We offer two examples of explicit calculations. As the first example, we determine the homotopy type of the independence complex of $L_m [H]$, where $L_m$ is the tree on $m$ vertices with no branches, for any positive integer $m$ when the independence complex of $H$ is homotopy equivalent to a wedge sum of $n$ copies of $d$-dimensional sphere. As the second one, for a forest $G$ and a complete graph $K$, we describe the homological connectivity of the independence complex of $G[K]$ by the independent domination number of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-09T11:37:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C76",
      "55P15"
    ],
    "keywords": [
      "independence complex",
      "lexicographic product",
      "homotopy equivalent",
      "wedge sum",
      "independent domination number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}