Compositions for matroids with the Fulkerson property (Q1900139)

A (finite, binary) matroid has the Fulkerson property with respect to a fixed element \(e\) if the minimal covering of the ground set by circuits that contain the element \(e\) is integral, for every non-negative objective function. Such a matroid is necessarily binary, and \textit{R. E. Bixby} [On the length-width inequality for compound clutters, J. Comb. Theory, Ser. B 11, 246-248 (1971; Zbl 0362.05026)] has given necessary and sufficient conditions under which a 2-sum of matroids has the Fulkerson property. In the current paper Chopra gives sufficient, but not necessary, conditions under which a 3-sum has the Fulkerson property. A characterization of such matroids in terms of excluded minors remains open. For the matroids which satisfy the stronger min-flow max-cut property \textit{P. D. Seymour} had given a complete characterization [The matroids with the max-flow min-cut property, J. Comb. Theory, Ser. B 23, 189-222 (1977; Zbl 0375.05022)].