关于内$4$连通二元拟阵VI的分裂定理
Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VI
摘要：
设$M$是一个$3$连通的二元拟阵;如果每个$3$ -分离的一条边是三角形或三元组，则$M$称为内部$4$ -连通的;如果每个$3$ -分离的一条边是三角形、三元组或$4$ -元扇，则$M$称为$(4,4,S)$ -连通的.假设$M$是内部$4$连通的，并且$M$及其对偶都不是三次莫比乌斯或平面梯或其某种余扩张。让$N$成为$M$的内部$4$连接的真小调。我们的目的是表明，$M$有一个适当的内部$4$连接的子与$N$ -子，可以从$M$通过删除最多四个元素，或通过删除元素在一个容易描述的方式从一个特殊的子结构的$M$。当这个目标不能满足时，本系列的早期论文表明，直到对偶，$M$有一个好的蝴蝶结，即一对$\{x_1,x_2,x_3\}$和$\{x_4,x_5,x_6\}$，不相交的三角形和一个上回路$\{x_2,x_3,x_4,x_5\}$，其中$M\backslash x_3$有一个$N$ -子式并且是$(4,4,S)$ -连通的。我们还表明，当$M$有一个好的蝴蝶结时，要么$M\backslash x_3,x_6$有一个$N$ -子式，要么$M\backslash x_3/x_2$有一个$N$ -子式，并且是$(4,4,S)$ -连通的。在本文中，我们表明，当$M\backslash x_3,x_6$有一个$N$ -子式，但不是$(4,4,S)$ -连接，$M$有一个内部$4$ -连接的正常子式与$N$ -子式，可以从$M$通过删除最多三个元素，或通过删除元素在一个良好的描述方式从$M$的几个特殊的子结构之一。这是一个重要的一步获得分裂定理类内部$4$连接的二元拟阵。
英文摘要：
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 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 $(4,4,S)$-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.
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