Structural properties of matroid matchings

This paper refines Lovasz’s duality theory for the linear matroid parity problem by: (1)characterizing a minimum cover in terms of maximum matchings, (2)characterizing maximum matchings in terms of a minimum cover and (3)characterizing critical structures called hypomatchable components. It describes a naturally arising lattice of minimum covers for primitive parity problems and characterizes the least and greatest elements in this lattice. For not necessarily primitive parity problems, the paper introduces a class of minimum covers whose members form a lattice and shows that the critical components in the least element of this lattice exhibit a special property called ‘hyper-matchability’.