{
  "id": "0812.1064",
  "version": "v1",
  "published": "2008-12-05T02:14:21.000Z",
  "updated": "2008-12-05T02:14:21.000Z",
  "title": "Graph Minors and Minimum Degree",
  "authors": [
    "Gašper Fijavž",
    "David R. Wood"
  ],
  "journal": "Electronic J. Combinatorics R151, 2010",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\\widehat{\\mathcal{D}}_k$. This paper discusses $\\widehat{\\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\\widehat{\\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\\widehat{\\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\\widehat{\\mathcal{D}}_k$ by adding one new vertex. For $k\\leq 3$ every graph in $\\widehat{\\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\\widehat{\\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\\widehat{\\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-05T02:14:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C83",
      "05C40"
    ],
    "keywords": [
      "minimum degree",
      "robertson-seymour graph minor theorem",
      "minor-minimal forbidden graphs",
      "complete multipartite graphs",
      "characterise"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.1064F"
    }
  }
}