Elastic elements in 3-connected matroids

TL;DR

This paper investigates the existence of 'elastic' elements in 3-connected matroids, showing that under certain conditions, such matroids have at least four elastic elements, expanding understanding of their structural properties.

Contribution

The paper introduces the concept of elastic elements in 3-connected matroids and proves their abundance under specific structural constraints.

Findings

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

Elastic elements are absent only in matroids with specific fan and 3-separating set structures.

Provides conditions under which elastic elements are guaranteed to exist.

Abstract

It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either ${\rm co}(M\delete e)$, the cosimplification of $M\delete e$, or ${\rm si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both ${\rm co}(M\delete e)$ and ${\rm si}(M/e)$ are $3$-connected. Calling such an element "elastic", in this paper we show that if $|E(M)|\ge 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and, up to duality, $M$ has no $3$-separating set $S$ that is the disjoint union of a rank-$2$ subset and a corank-$2$ subset of $E(M)$ such that $M|S$ is isomorphic to a member or a single-element deletion of a member of a certain family of matroids.