A decomposition theory for matroids. VI. Almost regular matroids

AbstractWe introduce a new class of binary matroids called almost regular. Any such matroid is not regular, but for any element, at least one of the two minors produced by deletion and contraction of that element must be regular. Furthermore, certain labels are assigned to the elements, and these labels must obey several conditions. In this part we prove that the entire class of almost regular matroids is producible from just two matroids by repeated application of elementary operations, each of which is a series expansion, or a parallel addition, or a substitution of a triangle by a triad, or a substitution of a triad by a triangle. In Part VII it will be shown that this result leads to new and surprisingly simple constructions for several matrix classes, in particular for the class of minimal violation matrices of total unimodularity. Up to now, no complete construction other than enumeration has been known for any of these classes