{
  "id": "2203.12397",
  "version": "v1",
  "published": "2022-03-23T13:17:56.000Z",
  "updated": "2022-03-23T13:17:56.000Z",
  "title": "On independent domination in direct products",
  "authors": [
    "Kirsti Kuenzel",
    "Douglas F. Rall"
  ],
  "comment": "14 pages, 2 figures, 2 tables",
  "categories": [
    "math.CO"
  ],
  "abstract": "In \\cite{nr-1996} Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that $i(G\\times H) \\ge i(G)i(H)$ where $i(G)$ is the independent domination number of $G$ and $G\\times H$ is the direct product of graphs $G$ and $H$. We show this conjecture is false, and, in fact, construct pairs of graphs for which $\\min\\{i(G), i(H)\\} - i(G\\times H)$ is arbitrarily large. We also give the exact value of $i(G\\times K_n)$ when $G$ is either a path or a cycle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-23T13:17:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C76"
    ],
    "keywords": [
      "direct product",
      "independent domination number",
      "graph products",
      "conjecture",
      "construct pairs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}