{
  "id": "0705.2439",
  "version": "v1",
  "published": "2007-05-16T21:23:59.000Z",
  "updated": "2007-05-16T21:23:59.000Z",
  "title": "A tight bound on the collection of edges in MSTs of induced subgraphs",
  "authors": [
    "Gregory B. Sorkin",
    "Angelika Steger",
    "Rico Zenklusen"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\\in\\{1,2,...,n-1\\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$ vertices has at most $nk-\\binom{k+1}{2}$ elements. This proves a conjecture of Goemans and Vondrak \\cite{GV2005}. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal $k$-connected graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-16T21:23:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C40",
      "05C05"
    ],
    "keywords": [
      "induced subgraphs",
      "tight bound",
      "collection",
      "distinct positive edge weights",
      "vertex graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.2439S"
    }
  }
}