{
  "id": "1609.07345",
  "version": "v1",
  "published": "2016-09-23T13:04:55.000Z",
  "updated": "2016-09-23T13:04:55.000Z",
  "title": "Stabilizing on the distinguishing number of a graph",
  "authors": [
    "Saeid Alikhani",
    "Samaneh Soltani"
  ],
  "comment": "10 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing stability, of a graph $G$ is denoted by $st_D(G)$ and is the minimum number of vertices whose removal changes the distinguishing number. We obtain a general upper bound $st_D(G) \\leqslant \\vert V(G)\\vert -D(G)+1$, and a relationships between the distinguishing stabilities of graphs $G$ and $G-v$, i.e., $st_D(G)\\leqslant st_D(G-v)+1$, where $v\\in V(G)$. Also we study the edge distinguishing stability number (distinguishing bondage number) of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-23T13:04:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "distinguishing number",
      "edge distinguishing stability number",
      "general upper bound",
      "stabilizing",
      "trivial automorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}