Quasiregular Matroids

Abstract

Regular matroids are binary matroids with no minors isomorphic to the Fano matroid $F_7$ or its dual $F_7^*$. Seymour proved that 3-connected regular matroids are either graphs, cographs, or $R_{10}$, or else can be decomposed along a non-minimal exact 3-separation induced by $R_{12}$. Quasiregular matroids are binary matroids with no minor isomorphic to the self-dual binary matroid $E_4$. The class of quasiregular matroids properly contains the class of regular matroids. We prove that 3-connected quasiregular matroids are either graphs, cographs, or deletion-minors of $PG(3,2)$, $R_{17}$ or $M_{12}$ or else can be decomposed along a non-minimal exact 3-separation induced by $R_{12}$, $P_9$, or $P_9^*$.