The internally 4-connected binary matroids with no M(K5\e)-minor

TL;DR

This paper classifies a specific class of highly connected binary matroids that exclude a certain minor, showing they are all related to a particular constructed matroid.

Contribution

It proves that all internally 4-connected binary matroids without an M(K5-e) minor are minors of a uniquely constructed matroid.

Findings

Characterization of internally 4-connected binary matroids without M(K5-e) minors

Identification of a universal matroid for this class

Structural insight into binary matroid minors

Abstract

Let AG(3,2)xU(1,1) denote the binary matroid obtained from the direct sum of AG(3,2) and a coloop by completing the 3-point lines between every element in AG(3,2) and the coloop. We prove that every internally 4-connected binary matroid that does not have a minor isomorphic to M(K5\e) is isomorphic to a minor of (AG(3,2)xU(1,1))*.