Detachable pairs in $3$-connected matroids and simple $3$-connected graphs

TL;DR

This paper characterizes when large 3-connected matroids and simple 3-connected graphs have detachable pairs, focusing on cases with 3-element circuits or cocircuits, extending previous results by Williams (2015).

Contribution

It provides a characterization of detachable pairs in 3-connected matroids with 13 or more elements, especially when they contain 3-element circuits or cocircuits, and applies this to simple 3-connected graphs.

Findings

Identifies conditions for the existence of detachable pairs in large 3-connected matroids.

Extends Williams' (2015) results to cases with 3-element circuits or cocircuits.

Characterizes when simple 3-connected graphs with at least 13 edges have detachable edge pairs.

Abstract

Let $M$ be a $3$-connected matroid. A pair $\{e,f\}$ in $M$ is detachable if $M \backslash e \backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following holds: $M$ has a detachable pair, $M$ has a $3$-element circuit or cocircuit, or $M$ is a spike. We address the case where $M$ has a $3$-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple $3$-connected graph $G$ with $|E(G)| \ge 13$ has a pair of edges $\{e,f\}$ such that $G/e/f$ or $G \backslash e\backslash f$ is simple and $3$-connected.