{
  "id": "1907.03158",
  "version": "v1",
  "published": "2019-07-06T17:18:14.000Z",
  "updated": "2019-07-06T17:18:14.000Z",
  "title": "$γ$-Graphs of Trees",
  "authors": [
    "Stephen Finbow",
    "Christopher M. van Bommel"
  ],
  "comment": "22 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G = (V, E)$, the $\\gamma$-graph of $G$, denoted $G(\\gamma) = (V(\\gamma), E(\\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\\gamma$-sets of $G$, and two $\\gamma$-sets are adjacent in $G(\\gamma)$ if they differ by a single vertex and the two different vertices are adjacent in $G$. In this paper, we consider $\\gamma$-graphs of trees. We develop an algorithm for determining the $\\gamma$-graph of a tree, characterize which trees are $\\gamma$-graphs of trees, and further comment on the structure of $\\gamma$-graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-06T17:18:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "cartesian product graphs",
      "vertex set",
      "single vertex",
      "minimum dominating sets",
      "connections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}