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

TL;DR

This paper advances the theory of internally 4-connected binary matroids by establishing conditions under which such matroids have minors related to a given minor, using element removal or specific substructure modifications.

Contribution

It proves a splitter theorem for internally 4-connected binary matroids, detailing how to obtain minors with a given minor through minimal element removals or structured substructure modifications.

Findings

If certain conditions are met, a minor with the desired properties exists after removing at most three elements.

The paper characterizes special substructures that facilitate obtaining the minor.

It extends previous results by handling cases where specific configurations are absent.

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.