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

TL;DR

This paper extends splitter theorems for 3-connected matroids by focusing on elastic elements, showing their structural properties and resolving related questions in matroid theory.

Contribution

It introduces a new splitter theorem based on elastic elements, generalizing previous results and linking elastic elements to path-width and minor preservation.

Findings

At least four elastic elements exist in certain 3-connected matroids.

Matroids with exactly four elastic elements have path-width three.

Resolves a question of Whittle and Williams about element removal relative to fixed bases.

Abstract

An element $e$ of a $3$-connected matroid $M$ is elastic if ${\rm si}(M/e)$, the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the cosimplification of $M\backslash e$, are both $3$-connected. It was recently shown that if $|E(M)|\geq 4$, 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.