Some well-known properties of the relaxation operation in matroid theory

Abstract

This thesis reviews the basic definitions and operations of matroid theory. In particular, the independent sets, circuits, bases, and rank of a vector matroid and a graphic matroid are described. All graphic matroids with a circuit-hyperplane on a ground set of up to six elements are listed. The interplay between circuit-hyperplane relaxation and other matroid operations is investigated. Lastly, the concept of a distinguishing set is used to prove a result characterizing matroids M and N on a common ground set with the property that, for each element e of the ground set, M/e = N/e or M \ e = N \ e.