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arXiv · 1804.06588

$N$-detachable pairs in 3-connected matroids III: the theorem

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 third and final paper in a series where we prove that if $|E(M)|-|E(N)| \ge 10$, then either $M$ has an $N$-detachable pair after possibly performing a single $\Delta$-$Y$ or $Y$-$\Delta$ exchange, or $M$ is essentially $N$ with a spike attached. Moreover, we describe the additional structures that arise if we require only that $|E(M)|-|E(N)| \ge 5$.

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Nick Brettell, Geoff Whittle, Alan Williams. 2018-04-18. $N$-detachable pairs in 3-connected matroids III: the theorem. https://doi.org/10.1016/j.jctb.2021.08.001

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