{
  "id": "2010.14273",
  "version": "v1",
  "published": "2020-10-27T13:14:23.000Z",
  "updated": "2020-10-27T13:14:23.000Z",
  "title": "$1/2$-conjectures on the domination game and claw-free graphs",
  "authors": [
    "Csilla Bujtás",
    "Vesna Iršič",
    "Sandi Klavžar"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\gamma_g(G)$ be the game domination number of a graph $G$. Rall conjectured that if $G$ is a traceable graph, then $\\gamma_g(G) \\le \\left\\lceil \\frac{1}{2}n(G)\\right\\rceil$. Our main result verifies the conjecture over the class of line graphs. Moreover, in this paper we put forward the conjecture that if $\\delta(G) \\geq 2$, then $\\gamma_g(G) \\leq \\left\\lceil \\frac{1}{2}n(G) \\right\\rceil$. We show that both conjectures hold true for claw-free cubic graphs. We further prove the upper bound $\\gamma_g(G) \\le \\left\\lceil \\frac{11}{20} \\, n(G) \\right\\rceil$ over the class of claw-free graphs of minimum degree at least $2$. Computer experiments supporting the new conjecture and sharpness examples are also presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-27T13:14:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C57",
      "05C69",
      "05C76"
    ],
    "keywords": [
      "claw-free graphs",
      "domination game",
      "game domination number",
      "claw-free cubic graphs",
      "conjectures hold true"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}