Splitters and Decomposers for Binary 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 $EX[S_{10},S_{10}^{\ast }]$, 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 $EX[M(K_{3,3}),M^{\ast }(K_{3,3}),M(K_5),M^{\ast }(K_5)]$ and $EX[M(K_{3,3}),M^{\ast }(K_{3,3})]$. 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}].

arXiv admin note: text overlap with arXiv:1403.7757