{
  "id": "2007.15921",
  "version": "v1",
  "published": "2020-07-31T09:30:52.000Z",
  "updated": "2020-07-31T09:30:52.000Z",
  "title": "The Localization Game On Cartesian Products",
  "authors": [
    "Jeandré Boshoff",
    "Adriana Roux"
  ],
  "comment": "16 pages, 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The localization game is played by two players: a Cop with a team of $k$ cops, and a Robber. The game is initialised by the Robber choosing a vertex $r \\in V$, unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes $k$ vertices and in return receives a distance vector. If the Cop can determine the exact location of $r$ from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at $r$, or move to $r'$ in the neighbourhood of $r$. The Cop then again probes $k$ vertices. The game continues in this fashion, where the Cop wins if the Robber can be located in a finite number of turns. The localization number $\\zeta(G)$, is defined as the least positive integer $k$ for which the Cop has a winning strategy irrespective of the moves of the Robber. In this paper, we focus on the game played on Cartesian products. We prove that $\\zeta( G \\square H) \\geq \\max\\{\\zeta(G), \\zeta(H)\\}$ as well as $\\zeta(G \\square H) \\leq \\zeta(G) + \\psi(H) - 1$ where $\\psi(H)$ is a doubly resolving set of $H$. We also show that $\\zeta(C_m \\square C_n)$ is mostly equal to two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-31T09:30:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C57",
      "05C76"
    ],
    "keywords": [
      "cartesian products",
      "localization game",
      "cop wins",
      "game proceeds turn",
      "cop probes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}