Applications of the Strong Splitter Theorem: decomposition results

TL;DR

This paper employs the Strong Splitter Theorem to decompose and classify binary matroids without an $E_4$-minor, identifying extremal and important matroids within this class.

Contribution

It introduces a decomposition framework for this class of matroids using the Strong Splitter Theorem, highlighting extremal and structurally significant matroids.

Findings

Identifies extremal matroids such as binary spikes and projective geometries.

Provides decomposition results for classes excluding $P_9$ and $P_9^*$.

Characterizes 3-connected members with specific 3-decomposers.

Abstract

We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no $E_4$-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important matroids in the class. The matroid $E_4$ is a self-dual 10-element binary 3-connected matroid that plays a useful role in structural results. It is a single-element coextension of $P_9$, which is a single-element extension of the 4-wheel. We show that the extremal matroids in this class are the binary rank-$r$ spikes $Z_r$, the rank 3 and 4 projective geometries $F_7$ and $PG(3,2)$, respectively, the 17-element internally 4-connected matroid $R_{17}$, and one 12-element rank-6 matroid. All the other 3-connected members have $P_9$ or $P_9^*$ as 3-decomposers. As immediate corollaries we get decomposition results for $EX[P_9^*]$ and $EX[P_9]$ as well as the internally 4-connected members of these classes.