{
  "id": "1607.04071",
  "version": "v1",
  "published": "2016-07-14T10:26:34.000Z",
  "updated": "2016-07-14T10:26:34.000Z",
  "title": "On the zero forcing number of corona and lexicographic product of graphs",
  "authors": [
    "I. Javaid",
    "I. Irshad",
    "M. Batool",
    "Z. Raza"
  ],
  "comment": "16 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S$ of black vertices (where vertices in $V(G)\\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications of $\"$the color change rule$\"$: a white vertex is turned black if it is the only white neighbor of a black vertex. In this paper, we study the zero forcing number of corona product, $G\\odot H$ and lexicographic product, $G\\circ H$ of two graphs $G$ and $H$. It is shown that if $G$ and $H$ are connected graphs of order $n_{1}\\geq2$ and $n_{2}\\geq2$ respectively, then $Z(G\\odot ^{k}H)=Z(G\\odot ^{k-1}H)+n_{1}(n_{2}+1)^{k-1}Z(H)$, where $G\\odot^{k}H=(G\\odot^{k-1}H)\\odot H$. Also, it is shown that for a connected graph $G$ of order $n\\geq 2$ and an arbitrary graph $H$ containing $l\\geq 1$ components $H_{1},H_{2}, \\cdots,H_{l}$ with $|V(H_{i})|=m_{i}\\geq 2$, $1\\leq i\\leq l$, $(n-1)l+\\sum\\limits_{i=1}^l m_{i}\\leq Z(G\\circ H)\\leq n(\\sum\\limits_{i=1}^{l}m_{i})-l$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-14T10:26:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "zero forcing number",
      "lexicographic product",
      "black vertex",
      "turned black",
      "color change rule"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}