Excluded minors are almost fragile II: essential elements

TL;DR

This paper advances the understanding of excluded minors in matroid theory by showing that certain large, non-fragile minors are close to having many $N$-essential elements, deepening the structural insights into $P$-representable matroids.

Contribution

It proves that under mild conditions, a large non-$N$-fragile minor is near a minor with many $N$-essential elements, extending previous results on the structure of excluded minors.

Findings

Large minors are close to having many $N$-essential elements.

$M ackslash a,b$ is one element away from a minor with at least $r(M)-2$ $N$-essential elements.

The results deepen the structural understanding of excluded minors for $P$-representable matroids.

Abstract

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb{P}$, let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary, and suppose $M$ has a pair of elements $\{a,b\}$ such that $M\backslash a,b$ is $3$-connected with an $N$-minor. Suppose also that $|E(M)| \geq |E(N)|+11$ and $M \backslash a,b$ is not $N$-fragile. In the prequel to this paper, we proved that $M \backslash a,b$ is at most five elements away from an $N$-fragile minor. An element $e$ in a matroid $M'$ is $N$-essential if neither $M'/e$ nor $M' \backslash e$ has an $N$-minor. In this paper, we prove that, under mild assumptions, $M \backslash a,b$ is one element away from a minor having at least $r(M)-2$ elements that are $N$-essential.