Computing excluded minors for classes of matroids representable over partial fields

TL;DR

This paper presents a computational approach to identify small excluded minors for classes of matroids representable over partial fields, specifically enumerating those with up to 15 elements for dyadic and 2-regular matroids, and discovering a 16-element minor for dyadic matroids.

Contribution

It introduces a computer search method to find excluded minors in matroid classes over partial fields, providing new enumerations and conjectures.

Findings

Enumerated excluded minors up to 15 elements for dyadic and 2-regular matroids

Discovered a 16-element excluded minor for dyadic matroids

Conjecture no other excluded minors exist for 2-regular matroids

Abstract

We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.