A characterization of certain excluded-minor classes of matroids (Q1263591)

\textit{P. N. Walton} [``Some topics in combinatorial theory'', D. Phil. Thesis, 1981] and \textit{J. G. Oxley} [``On non-binary 3-connected matroids'', Trans. Am. Math. Soc. 300, 663-679 (1987; Zbl 0628.05016)] showed that every 3-connected matroid of rank and corank at least 3 has one of five six-element matroids as minor each of which has rank-3 and is self-dual. The author characterizes two classes of matroids that arise when one excludes only three of these minors. The first class leads to an extension of a characterization of ternary matroids with no \(M(K_ 4)\) minor given in \textit{J. G. Oxley} [``A characterization of the ternary matroids with no \(M(K_ 4)\) minor'', J. Comb. Theory, Ser. B 42, 212-249 (1987; Zbl 0585.05005)] and the second class generalizes the excluded- minor characterization of \textit{W. T. Tutte} [``Connectivity in matroids'', Can. J. Math. 18, 1301-1324 (1966; Zbl 0149.215)].