Abstract

We consider the $\mathrm{GF}(4)$-representable matroids with a circuit-hyperplane such that the matroid obtained by relaxing the circuit-hyperplane is also $\mathrm{GF}(4)$-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 $\mathrm{GF}(4)$-representations of these matroids.