A decomposition theory for matroids. VI: Almost regular matroids
This paper introduces the class of almost regular matroids. These are the binary matroids \(M\) for which there is a partition (del, con) of \(E(M)\) into non-empty sets such that (i) \(M\backslash e\) is regular if \(e\) is in del; (ii) \(M/e\) is regular if \(e\) is in con; and (iii) every circuit contains an even number of members of con while every cocircuit contains an even number of members of del. The author shows that if \(M\) is such a matroid, then \(M\) is connected, its dual is almost regular, and del is a circuit-hyperplane. The main result of the paper is that, by a sequence of series of parallel extensions and \(\Delta Y\) or \(Y\Delta\) operations, every almost regular matroid can be constructed from either the Fano matroid, \(F_ 7\), or a certain single-element extension of \(R_{10}\).
scientific article; zbMATH DE number 130570