# Relaxations of GF$(4)$-representable matroids

## TL;DR

This paper characterizes GF(4)-representable matroids with a special circuit-hyperplane that remains GF(4)-representable after relaxation, using structure theorems and identifying forbidden submatrices.

## Contribution

It provides a structural characterization of GF(4)-representable matroids with relaxable circuit-hyperplanes and identifies forbidden submatrices in their representations.

## Key findings

- Structural characterization of GF(4)-representable matroids with relaxable circuit-hyperplanes.
- Identification of forbidden submatrices in GF(4)-representations.
- Application of structure theorems for fragile matroids.

## Abstract

We consider the GF$(4)$-representable matroids with a circuit-hyperplane such that the matroid obtained by relaxing the circuit-hyperplane is also GF$(4)$-representable. We characterize the structure of these matroids as an application of structure theorems for the classes of $U_{2,4}$-fragile and $\{U_{2,5},U_{3,5}\}$-fragile matroids. In addition, we characterize the forbidden submatrices in GF$(4)$-representations of these matroids.