{
  "id": "1309.1926",
  "version": "v1",
  "published": "2013-09-08T06:06:05.000Z",
  "updated": "2013-09-08T06:06:05.000Z",
  "title": "On graphs with no induced subdivision of $K_4$",
  "authors": [
    "Benjamin Lévêque",
    "Frédéric Maffray",
    "Nicolas Trotignon"
  ],
  "journal": "Journal of Combinatorial Theory, Series B, 102:924-947, 2012",
  "doi": "10.1016/j.jctb.2012.04.005",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove a decomposition theorem for graphs that do not contain a subdivision of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four vertices. We obtain also a structure theorem for the class $\\cal C$ of graphs that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph, where a wheel is a cycle on at least four vertices together with a vertex that has at least three neighbors on the cycle. Our structure theorem is used to prove that every graph in $\\cal C$ is 3-colorable and entails a polynomial-time recognition algorithm for membership in $\\cal C$. As an intermediate result, we prove a structure theorem for the graphs whose cycles are all chordless.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-08T06:06:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "induced subdivision",
      "structure theorem",
      "induced subgraph",
      "polynomial-time recognition algorithm",
      "decomposition theorem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.1926L"
    }
  }
}