  If S is a set of matroids, then the matroid M is S-fragile if, for every
element e in E(M), either M\e or M/e has no minor isomorphic to a member of S.
Excluded-minor characterizations often depend, implicitly or explicitly, on
understanding classes of fragile matroids. In certain cases, when F is a
minor-closed class of S-fragile matroids, and N is in F, the only members of F
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.
