Fan-Extensions in Fragile Matroids

Abstract

If $\mathcal{S}$ is a set of matroids, then the matroid $M$ is $\mathcal{S}$-fragile if, for every element $e\in E(M)$, either $M\backslash e$ or $M/e$ has no minor isomorphic to a member of $\mathcal{S}$. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when $\mathcal{M}$ is a minor-closed class of $\mathcal{S}$-fragile matroids, and $N\in \mathcal{M}$, the only members of $\mathcal{M}$ that contain $N$ as a minor are obtained from $N$ by increasing the length of fans. We prove that if this is the case, then we can certify it with a finite case-analysis. The analysis  involves examining matroids that are at most two elements larger than $N$.