{
  "id": "1708.02414",
  "version": "v1",
  "published": "2017-08-08T09:16:52.000Z",
  "updated": "2017-08-08T09:16:52.000Z",
  "title": "Strong geodetic problem on Cartesian products of graphs",
  "authors": [
    "Vesna Iršič",
    "Sandi Klavžar"
  ],
  "comment": "18 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest path between a pair of vertices from $S$. In this paper, the strong geodetic problem is studied on the Cartesian product of graphs. A general upper bound for ${\\rm sg}(G \\,\\square\\, H)$ is determined, as well as exact values for $K_m \\,\\square\\, K_n$, $K_{1, k} \\,\\square\\, P_l$, and certain prisms. Connections between the strong geodetic number of a graph and its subgraphs are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-08T09:16:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C70",
      "05C76",
      "68Q17"
    ],
    "keywords": [
      "strong geodetic problem",
      "cartesian product",
      "strong geodetic number",
      "smallest vertex subset",
      "general upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}