A splitter theorem for elastic elements in $3$-connected matroids

Abstract

An element e of a 3-connected matroid M is elastic if si(M/e), the simplification of M/e, and co(M \e), the cosimplification of M \e, are both 3-connected. It was recently shown that if |E(M )| ≥ 4, then M has at least four elastic elements provided M has no 4-element fans and no member of a specific family of 3-separators. In this paper, we extend this wheels-and-whirls type result to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified 3-connected minor. We also prove that if M has exactly four elastic elements, then it has path-width three. Lastly, we resolve a question of Whittle and Williams, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of this work.