Splitters and Decomposers for Binary Matroids

TL;DR

This paper develops a decomposition theorem for certain binary matroids excluding specific minors, leading to new structural insights and implications for internally 4-connected matroids.

Contribution

It introduces a novel decomposition theorem for the class excluding the matroid $S_{10}$ and its dual, extending to classes excluding Kuratowski graphs, advancing structural understanding of binary matroids.

Findings

Decomposition theorem for $EX[S_{10}, S_{10}^*]$.

Corollaries for classes excluding Kuratowski minors.

Implications for internally 4-connected matroids.

Abstract

Let $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As corollaries we obtain decomposition theorems for the classes obtained by excluding the Kuratowski graphs $EX[M(K_{3,3}), M^*(K_{3,3}), M(K_5), M^*(K_5)]$ and $EX[M(K_{3,3}), M^*(K_{3,3})]$. These decomposition theorems imply results on internally $4$-connected matroids by Zhou [\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte [\ref{Mayhewsubmitted}].