Excluded minors are almost fragile

TL;DR

The paper investigates the structure of excluded minors in matroid theory, showing they are nearly fragile with respect to certain stabilizers, which advances understanding of matroid representability and minor-closed classes.

Contribution

It proves that excluded minors are either bounded or can be transformed to exhibit fragility relative to a stabilizer, revealing near-fragility properties in matroid minors.

Findings

Excluded minors are either bounded or can be made fragile.

A pair of elements can be chosen to exhibit fragility after certain transformations.

The results connect excluded minors with matroid fragility and stabilizer properties.

Abstract

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded relative to $N$, or, up to replacing $M$ by a $\Delta$-$Y$-equivalent excluded minor, we can choose a pair of elements $\{a,b\}$ such that either $M\backslash \{a,b\}$ is $N$-fragile, or $M^* \backslash \{a,b\}$ is $N^*$-fragile.