Quasiregular matroids (Q1658769)

Summary: Regular matroids are binary matroids with no minors isomorphic to the Fano matroid \(F_7\) or its dual \(F_7^\ast\). \textit{P. D. Seymour} [J. Comb. Theory, Ser. B 28, 305--359 (1980; Zbl 0443.05027)] 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 \(\mathrm{PG}(3,2)\), \(R_{17}\) or \(M_{12}\) or else can be decomposed along a non-minimal exact 3-separation induced by \(R_{12}\), \(P_9\), or \(P_9^\ast\).