Matroid Cocircuit

A matroid cocircuit is a matroid circuit of the dual matroid of a matroid. The dual matroid has as its matroid bases the complements, in the ground set, of the original matroid bases. Equivalently, a matroid cocircuit is an inclusion-minimal set meeting every matroid basis. A matroid circuit and a matroid cocircuit cannot intersect in exactly one element (Fu 2026).

Every simple matroid whose matroid cocircuits have at least four elements contains one of nine specified matroid minors, each of matroid rank at most 5 with at most twelve elements (Mizell and Oxley 2026).