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arXiv · 1804.06029

$N$-detachable pairs in 3-connected matroids II: life in $X$

Nick Brettell·Geoff Whittle·Alan Williams
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Abstract

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the second in a series of three papers where we describe the structures that arise when it is not possible to find an $N$-detachable pair in $M$. In the first paper in the series, we showed that, under mild assumptions, either $M$ has an $N$-detachable pair, $M$ has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs, or there is a 3-separating set $X$ with certain strong structural properties. In this paper, we analyse matroids with such a structured set $X$, and prove that they have either an $N$-detachable pair, or one of five particular 3-separators that can appear in a matroid with no $N$-detachable pairs.

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Nick Brettell, Geoff Whittle, Alan Williams. 2018-04-17. $N$-detachable pairs in 3-connected matroids II: life in $X$. https://doi.org/10.1016/j.jctb.2020.07.008

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