Structural properties of matroid matchings

AbstractThis paper refines Lovasz's duality theory for the linear matroid parity problem by: 1.(1) characterizing a minimum cover in terms of maximum matchings,2.(2) characterizing maximum matchings in terms of a minimum cover and3.(3) characterizing critical structures called hypomatchable components.We describe a naturally arising lattice of minimum covers for primitive parity problems and characterize the least and greatest elements in this lattice. For not necessarily primitive parity problems, we introduce a class of minimum covers whose members form a lattice and show that the critical components in the least element of this lattice exhibit a special property called “hypermatchability”