Classes of binary matroids with small lists of excluded induced minors

TL;DR

This paper extends previous work by characterizing classes of binary matroids that exclude certain small minors as induced minors, providing a comprehensive understanding of their structural properties.

Contribution

It determines classes of matroids avoiding pairs of specific minors and advances the structural characterization of matroids excluding a particular minor.

Findings

Classified matroids avoiding pairs of minors such as M(C4) and M(K4)

Established structural lemmas for matroids excluding M(K4) as an induced minor

Extended the framework for understanding excluded induced minors in binary matroids

Abstract

In earlier work, we characterized the class of matroids with no $M(C_4)$ as an induced minor and the class of matroids with no member of $\{M(C_4),M(K_4)\}$ as an induced minor. In this paper, for every two matroids in $\{M(C_4),M(K_4\backslash e),M(K_4),F_7\}$, we determine the class of matroids that have neither of the chosen pair as an induced minor. Additionally, we prove structural lemmas toward characterizing the class of matroids that do not contain $M(K_4)$ as an induced minor.