Excluded minors are almost fragile II: Essential elements

Abstract

Let M be an excluded minor for the class of $P$-representable matroids for some partial field $P$, let N be a 3-connected strong $P$-stabilizer that is non-binary, and suppose M has a pair of elements $\{a,b\}$ such that $M﹨a,b$ is 3-connected with an N-minor. Suppose also that $\left|E\right(M\left)\right|\ge \left|E\right(N\left)\right|+11$ and $M﹨a,b$ is not N-fragile. In the prequel to this paper, we proved that $M﹨a,b$ 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 $M^{\prime }﹨e$ has an N-minor. In this paper, we prove that, under mild assumptions, $M﹨a,b$ is one element away from a minor having at least $r\left(M\right)-2$ elements that are N-essential.