Matroid 3-connectivity and branch width

TL;DR

This paper proves that large branch width in 3-connected matroids guarantees the existence of a small subset whose removal or contraction preserves 3-connectivity and a specific minor.

Contribution

It establishes a new link between branch width and connectivity preservation in matroids with a given minor.

Findings

Large branch width ensures the existence of a small subset maintaining 3-connectivity.

The result applies to any nonnegative integer k and any matroid N as a minor.

The theorem generalizes previous connectivity and minor-preservation results in matroid theory.

Abstract

We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X and M/X is 3-connected and contains N as a minor.