Intertwining connectivities in representable matroids

TL;DR

This paper investigates how to modify representable matroids by removing elements while maintaining specific connectivities and minors, providing new insights into the structure and properties of large matroids.

Contribution

It establishes conditions under which elements can be removed from large representable matroids without losing key connectivities and minors, advancing matroid theory.

Findings

Existence of an element preserving both connectivities upon removal or contraction.

Stronger result for finite field representable matroids maintaining connectivity and minors.

Applicable to large matroids, extending structural understanding.

Abstract

Let $M$ be a representable matroid, and $Q, R, S, T$ subsets of the ground set. We prove that, if $M$ is sufficiently large, then there is an element $e$ such that deleting or contracting $e$ preserves both the $Q$-$R$ and the $S$-$T$ connectivities. For matroids representable over a finite field we prove a stronger result: we show that we can remove $e$ such that both a connectivity and a minor of $M$ are preserved.