{
  "id": "1605.07016",
  "version": "v1",
  "published": "2016-05-23T14:03:25.000Z",
  "updated": "2016-05-23T14:03:25.000Z",
  "title": "Distinguishing number and distinguishing index of some operations on graphs",
  "authors": [
    "Saeid Alikhani",
    "Samaneh Soltani"
  ],
  "comment": "11 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. We examine the effects on $D(G)$ and $D'(G)$ when $G$ is modified by operations on vertex and edge of $G$. Let $G$ be a connected graph of order $n\\geq 3$. We show that $-1\\leq D(G-v)-D(G)\\leq D(G)$, where $G-v$ denotes the graph obtained from $G$ by removal of a vertex $v$ and all edges incident to $v$ and these inequalities are true for the distinguishing index. Also we prove that $|D(G-e)-D(G)|\\leq 2$ and $-1 \\leq D'(G-e)-D'(G)\\leq 2$, where $G-e$ denotes the graph obtained from $G$ by simply removing the edge $e$. Finally we consider the vertex contraction and the edge contraction of $G$ and prove that the edge contraction decrease the distinguishing number (index) of $G$ by at most one and increase by at most $3D(G)$ ($3D'(G)$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-23T14:03:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05E18"
    ],
    "keywords": [
      "distinguishing number",
      "distinguishing index",
      "operations",
      "edge contraction decrease",
      "vertex contraction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}