The structure of $\{U_{2,5}, U_{3,5}\}$-fragile matroids

TL;DR

This paper provides a structural characterization of strictly U_{2,5},U_{3,5}ragile matroids with six GF(5) representations, distinguishing those with and without certain minors, and relates to excluded minors for specific matroid classes.

Contribution

It offers a detailed structural description of U_{2,5},U_{3,5}ragile matroids with six GF(5) representations, aiding in classifying excluded minors for key matroid classes.

Findings

Matroids without X_8, Y_8, Y_8^*ollows from two base matroids by gluing wheels.

Matroids with X_8, Y_8, Y_8^*eature path width 3 and are constructed via elementary operations.

Characterization aids in identifying excluded minors for Hydra-5 and 2-regular matroids.

Abstract

Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is strictly $\mathcal{N}$-fragile if $M$ has a member of $\mathcal{N}$ as minor and, for all $e \in E(M)$, at least one of $M\backslash e$ and $M/e$ has no minor in $\mathcal{N}$. In this paper we give a structural description of the strictly $\{U_{2,5},U_{3,5}\}$-fragile matroids that have six inequivalent representations over $\mathrm{GF}(5)$. Roughly speaking, these matroids fall into two classes. The matroids without an $\{X_8, Y_8, Y_8^{*}\}$-minor are constructed, up to duality, from one of two matroids by gluing wheels onto specified triangles. On the other hand, those matroids with an $\{X_8, Y_8, Y_8^{*}\}$-minor can be constructed from a matroid in $\{X_8, Y_8, Y_8^{*}\}$ by repeated application of elementary operations, and are shown to have path width 3. The characterization presented here will be crucial in finding the explicit list of excluded minors for two classes of matroids: the Hydra-5-representable matroids and the 2-regular matroids.