Excluded minors are almost fragile II: essential elements

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 $|E(M)|\ge |E(N)|+11$ 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 $M^{\prime }$ is $N$-essential if neither $M^{\prime }/e$ nor $M^{\prime }\backslash e$ 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.

34 pages, 2 figures

References in corpus (1)

• The excluded minors for 2- and 3-regular matroids

Cited by in corpus (1)

• Computing excluded minors for classes of matroids representable over partial fields