A note on the connectivity of 2-polymatroid minors

TL;DR

This paper extends a known connectivity property from matroids to 2-polymatroids, showing that element removal can preserve connectivity and minors, and explores the uniqueness of such element removals.

Contribution

It proves a weaker but analogous connectivity result for 2-polymatroids and investigates the uniqueness of element removal methods.

Findings

Existence of an element whose removal preserves connectivity and minors.

Extension of matroid connectivity results to 2-polymatroids.

Discussion on the uniqueness of element removal for maintaining connectivity.

Abstract

Brylawski and Seymour independently proved that if $M$ is a connected matroid with a connected minor $N$, and $e \in E(M) - E(N)$, then $M \backslash e$ or $M / e$ is connected having $N$ as a minor. This paper proves an analogous but somewhat weaker result for $2$-polymatroids. Specifically, if $M$ is a connected $2$-polymatroid with a proper connected minor $N$, then there is an element $e$ of $E(M) - E(N)$ such that $M \backslash e$ or $M / e$ is connected having $N$ as a minor. We also consider what can be said about the uniqueness of the way in which the elements of $E(M) - E(N)$ can be removed so that connectedness is always maintained.