If S is a set of matroids, then the matroid M is S-fragile if, for every element e ∈ 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 M is a minor-closed class of S-fragile matroids, and N ∈ M, the only members of 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 .
For all positive integers t exceeding one, a matroid has the cyclic (t − 1, t)-property if its ground set has a cyclic ordering σ such that every set of t − 1 consecutive elements in σ is contained in a telement circuit and t-element cocircuit. We show that if M has the cyclic (t − 1, t)-property and |E(M )| is sufficiently large, then these telement circuits and t-element cocircuits are arranged in a prescribed way in σ, which, for odd t, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even t, is analogous to how 4-element circuits and cocircuits appear in swirls. Furthermore, we show that any appropriate concatenation Φ of σ is a flower. If t is odd, then Φ is a daisy, but if t is even, then, depending on M , it is possible for Φ to be either an anemone or a daisy.
We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.
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