{ "id": "1608.01027", "version": "v1", "published": "2016-08-02T23:08:17.000Z", "updated": "2016-08-02T23:08:17.000Z", "title": "Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VII", "authors": [ "Carolyn Chun", "James Oxley" ], "comment": "55 pages, 29 figures", "categories": [ "math.CO" ], "abstract": "Let $M$ be a $3$-connected binary matroid; $M$ is 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 \\ffsc. We also showed that, when $M$ has a good bowtie, either $M\\backslash x_3,x_6$ has an $N$-minor and $M\\backslash x_6$ is $(4,4,S)$-connected; or $M\\backslash x_3/x_2$ has an $N$-minor and is \\ffsc. In this paper, we show that, when $M\\backslash x_3,x_6$ has no $N$-minor, $M$ 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 a special substructure of $M$. This is the penultimate step towards obtaining a splitter theorem for the class of internally $4$-connected binary matroids.", "revisions": [ { "version": "v1", "updated": "2016-08-02T23:08:17.000Z" } ], "analyses": { "subjects": [ "05B35", "05C40" ], "keywords": [ "connected binary matroids vii", "splitter theorem", "connected proper minor", "special substructure", "removing elements" ], "note": { "typesetting": "TeX", "pages": 55, "language": "en", "license": "arXiv", "status": "editable" } } }