On Matroid Connectivity.

Certain classes of 2- and 3-connected matroids are studied in this thesis. In Chapter 2 we give a characterization of those 2-connected matroids M with the property that, for a given positive integer m, the deletion of every non-empty subset of M having at most m elements is disconnected. A bound on the maximum number of elements of such a matroid in terms of its rank is also given, along with a complete description of the matroids attaining this bound. These results extend results of Murty and Oxley for minimally 2-connected matroids. A characterization of the 3-connected matroids M that have the property that every 2-element deletion of M is disconnected is given in Chapter 3. It is shown that these matroids are exactly the duals of Sylvester matroids having at least four elements. In Chapter 4 we prove the following result: Let M be a 3-connected matroid other than a wheel of rank greater than three, and let C be a circuit of M. If the deletion of every pair of elements of C is disconnected, then every pair of elements of C is contained in a triad of M. For an integer t greater than one, an n-element matroid M is t-cocyclic if every deletion having at least n $-$ t + 1 elements is 2-connected, and every deletion having exactly n $-$ t elements is disconnected. A matroid is t-cyclic if its dual is t-cocyclic. In Chapter 5 we investigate the matroids that are both t-cocyclic and t-cyclic. It is shown that these matroids are exactly the uniform matroids U (t,2t) and the Steiner Systems S(t, t + 1, 2t + 2)