The excluded minors for near-regular matroids

TL;DR

This paper provides a proof characterizing near-regular matroids as those avoiding a specific set of minors, clarifying the structural conditions for near-regularity in matroid theory.

Contribution

It offers a formal proof of the previously unpublished characterization of near-regular matroids by their excluded minors.

Findings

Near-regular matroids are characterized by specific excluded minors.

The set of excluded minors includes U2,5, U3,5, Fano plane, non-Fano, and others.

The proof solidifies the structural understanding of near-regular matroids.

Abstract

In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization.