{
  "id": "1608.01022",
  "version": "v1",
  "published": "2016-08-02T22:44:59.000Z",
  "updated": "2016-08-02T22:44:59.000Z",
  "title": "Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VI",
  "authors": [
    "Carolyn Chun",
    "James Oxley"
  ],
  "comment": "60 pages, 30 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $M$ be a $3$-connected binary matroid; $M$ is called internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a $4$-element fan. Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic M\\\"{o}bius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. Our aim is to show that $M$ has a proper internally $4$-connected minor with an $N$-minor that can be obtained from $M$ either by removing at most four elements, or by removing elements in an easily described way from a special substructure of $M$. When this aim cannot be met, the earlier papers in this series showed that, up to duality, $M$ has a good bowtie, that is, a pair, $\\{x_1,x_2,x_3\\}$ and $\\{x_4,x_5,x_6\\}$, of disjoint triangles and a cocircuit, $\\{x_2,x_3,x_4,x_5\\}$, where $M\\backslash x_3$ has an $N$-minor and is $(4,4,S)$-connected. We also showed that, when $M$ has a good bowtie, either $M\\backslash x_3,x_6$ has an $N$-minor; or $M\\backslash x_3/x_2$ has an $N$-minor and is $(4,4,S)$-connected. In this paper, we show that, when $M\\backslash x_3,x_6$ has an $N$-minor but is not $(4,4,S)$-connected, $M$ has an internally $4$-connected proper minor with an $N$-minor that can be obtained from $M$ by removing at most three elements, or by removing elements in a well-described way from one of several special substructures of $M$. This is a significant step towards obtaining a splitter theorem for the class of internally $4$-connected binary matroids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-02T22:44:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35",
      "05C40"
    ],
    "keywords": [
      "connected binary matroids vi",
      "splitter theorem",
      "connected proper minor",
      "special substructure",
      "removing elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 60,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}