Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VII
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 Möbius 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.