{
  "id": "2011.01763",
  "version": "v1",
  "published": "2020-11-03T15:04:32.000Z",
  "updated": "2020-11-03T15:04:32.000Z",
  "title": "A Bipartite Graph That Is Not the $γ$-Graph of a Bipartite Graph",
  "authors": [
    "Christopher M. van Bommel"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G = (V, E)$, the $\\gamma$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $\\gamma$-sets of $G$, and two $\\gamma$-sets are adjacent if they differ by a single vertex and the two different vertices are adjacent in $G$. An open question in $\\gamma$-graphs is whether every bipartite graph is the $\\gamma$-graph of some bipartite graph. We answer this question in the negative by demonstrating that $K_{2, 3}$ is not the $\\gamma$-graph of any bipartite graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-03T15:04:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "bipartite graph",
      "vertex set",
      "single vertex",
      "open question"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}