The excluded minors for 2- and 3-regular matroids

TL;DR

This paper characterizes the excluded minors for 2- and 3-regular matroids, revealing their structure, representation properties, and bounds on their size, thereby advancing the understanding of these matroid classes.

Contribution

It provides the first excluded-minor characterization for 2-regular matroids and bounds the size of excluded minors for 3-regular matroids.

Findings

Excluded minors for 2-regular matroids characterized.

3-regular matroids coincide with those representable over Hydra-5 partial field.

Excluded minors for 3-regular matroids have at most 15 elements.

Abstract

The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of matroids representable over the Hydra-5 partial field, and the 3-connected matroids in the class with a $U_{2,5}$- or $U_{3,5}$-minor are precisely those with six inequivalent representations over GF(5). We also prove that an excluded minor for this class has at most 15 elements.