{
  "id": "1601.02234",
  "version": "v1",
  "published": "2016-01-10T17:01:25.000Z",
  "updated": "2016-01-10T17:01:25.000Z",
  "title": "Hypo-efficient domination and hypo-unique domination",
  "authors": [
    "Vladimir Samodivkin"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G$ let $\\gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$\\mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets,but $G-v$ has a unique minimum dominating set for each $v\\in V(G)$. We show that each hypo-$\\mathcal{UD}$ graph $G$ of order at least $3$ is connected and $\\gamma(G-v) < \\gamma(G)$ for all $v \\in V(G)$. We obtain a tight upper bound on the order of a hypo-$\\mathcal{P}$ graph in terms of the domination number and maximum degree of the graph, where $\\mathcal{P} \\in \\{\\mathcal{UD}, \\mathcal{ED}\\}$. Families of circulant graphs which achieve these bounds are presented. We also prove that the bondage number of any hypo-$\\mathcal{UD}$ graph is not more than the minimum degree plus one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-10T17:01:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "domination number",
      "hypo-efficient domination graph",
      "hypo-unique domination graph",
      "tight upper bound",
      "unique minimum dominating set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102234S"
    }
  }
}