{
  "id": "0902.1121",
  "version": "v1",
  "published": "2009-02-06T19:10:29.000Z",
  "updated": "2009-02-06T19:10:29.000Z",
  "title": "A Linear-Time Algorithm for the Maximum Matched-Paired-Domination Problem in Cographs",
  "authors": [
    "Ruo-Wei Hung",
    "Chih-Chia Yao"
  ],
  "comment": "23 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G=(V,E)$ be a graph without isolated vertices. A matching in $G$ is a set of independent edges in $G$. A perfect matching $M$ in $G$ is a matching such that every vertex of $G$ is incident to an edge of $M$. A set $S\\subseteq V$ is a \\textit{paired-dominating set} of $G$ if every vertex in $V-S$ is adjacent to some vertex in $S$ and if the subgraph $G[S]$ induced by $S$ contains at least one perfect matching. The paired-domination problem is to find a paired-dominating set of $G$ with minimum cardinality. A set $MPD\\subseteq E$ is a \\textit{matched-paired-dominating set} of $G$ if $MPD$ is a perfect matching of $G[S]$ induced by a paired-dominating set $S$ of $G$. Note that the paired-domination problem can be regard as finding a matched-paired-dominating set of $G$ with minimum cardinality. Let $\\mathcal{R}$ be a subset of $V$, $MPD$ be a matched-paired-dominating set of $G$, and let $V(MPD)$ denote the set of vertices being incident to edges of $MPD$. A \\textit{maximum matched-paired-dominating set} $MMPD$ of $G$ w.r.t. $\\mathcal{R}$ is a matched-paired-dominating set such that $|V(MMPD)\\cap \\mathcal{R}|\\geqslant |V(MPD)\\cap \\mathcal{R}|$. An edge in $MPD$ is called \\textit{free-paired-edge} if neither of its both vertices is in $\\mathcal{R}$. Given a graph $G$ and a subset $\\mathcal{R}$ of vertices of $G$, the \\textit{maximum matched-paired-domination problem} is to find a maximum matched-paired-dominating set of $G$ with the least free-paired-edges; note that, if $\\mathcal{R}$ is empty, the stated problem coincides with the classical paired-domination problem. In this paper, we present a linear-time algorithm to solve the maximum matched-paired-domination problem in cographs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-06T19:10:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C85",
      "68Q25"
    ],
    "keywords": [
      "maximum matched-paired-domination problem",
      "linear-time algorithm",
      "matched-paired-dominating set",
      "perfect matching",
      "minimum cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.1121H"
    }
  }
}