On the intersections of circuits and cocircuits in matroids

Abstract

Seymour has shown that a matroid has a triad, that is, a 3-element set which is the intersection of a circuit and a cocircuit, if and only if it is non-binary. In this paper we determine precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit. We also show that this will occur ifM has a circuit and a cocircuit meeting in more than four elements. In addition, we prove that if a 3-connected matroid has a quad, then every pair of elements is in a quad. The corresponding result for triads was proved by Seymour.

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Oxley, J.G. On the intersections of circuits and cocircuits in matroids. Combinatorica 4, 187–195 (1984). https://doi.org/10.1007/BF02579220

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AMS subject classification (1980)

  • 05 B 35