{
  "id": "1807.04997",
  "version": "v1",
  "published": "2018-07-13T10:32:11.000Z",
  "updated": "2018-07-13T10:32:11.000Z",
  "title": "MAX for $k$-independence in multigraphs",
  "authors": [
    "Nevena Francetić",
    "Sara Herke",
    "Daniel Horsley"
  ],
  "comment": "16 pages, 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a fixed positive integer $k$, a set $S$ of vertices of a graph or multigraph is called a {\\it $k$-independent set} if the subgraph induced by $S$ has maximum degree less than $k$. The well-known algorithm MAX finds a maximal $k$-independent set in a graph or multigraph by iteratively removing vertices of maximum degree until what remains has maximum degree less than $k$. We give an efficient procedure that determines, for a given degree sequence $D$, the smallest cardinality $b(D)$ of a $k$-independent set that can result from any application of MAX to any loopless multigraph with degree sequence $D$. This analysis of the worst case is sharp for each degree sequence $D$ in that there exists a multigraph $G$ with degree sequence $D$ such that some application of MAX to $G$ will result in a $k$-independent set of cardinality exactly $b(D)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-13T10:32:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05B40"
    ],
    "keywords": [
      "degree sequence",
      "multigraph",
      "independent set",
      "maximum degree",
      "well-known algorithm max finds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}