On the intersections of circuits and cocircuits in matroids
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 matroid M has a quad, a 4-element set which is the intersection of a circuit and a cocircuit. We also show that this will occur if M 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. © 1984 Akadémiai Kiadó.
Publication Source (Journal or Book title)
Oxley, J. (1984). On the intersections of circuits and cocircuits in matroids. Combinatorica, 4 (2-3), 187-195. https://doi.org/10.1007/BF02579220