On two classes of nearly binary matroids

TL;DR

This paper characterizes two classes of matroids that are nearly binary, providing excluded-minor characterizations for each, and explores their relationship to binary matroids and circuit-hyperplane relaxations.

Contribution

It introduces new excluded-minor characterizations for two classes of nearly binary matroids, expanding understanding of their structure and relation to binary matroids.

Findings

Characterization of matroids where M or M/e is binary for all e

Excluded-minor characterization of these classes

Relation to matroids obtained from binary ones by relaxing a circuit-hyperplane

Abstract

We give an excluded-minor characterization for the class of matroids M in which M\e or M/e is binary for all e in E(M). This class is closely related to the class of matroids in which every member is binary or can be obtained from a binary matroid by relaxing a circuit-hyperplane. We also provide an excluded-minor characterization for the second class.