Reinforcing a Matroid to Have k Disjoint Bases

Let ( ) M  denote the maximum number of disjoint bases in a matroid M . For a connected graph G , let ( ) = ( ( )) G M G   , where ( ) M G is the cycle matroid of G . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs G with ( ) G k   . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid M with ( ) M k   , elements ( ) e E M  with the property that ( ) M e k    have been characterized in terms of matroid invariants such as strength and  -partitions. In this paper, we consider matroids M with ( ) < M k  , and determine the minimum of | ( ) | | ( ) | E M E M   , where M  is a matroid that contains M as a restriction with both ( ) = ( ) r M r M  and ( ) M k    . This minimum is expressed as a function of certain invariants of M , as well as a min-max formula. These are applied to imply former results of Haas [3] and of Liu et al. [4].