On packing minors into connected matroids

Let N be a matroid with k connected components and M be a minor-minimal connected matroid having N as a minor. This note proves that |E(M) − E(N)| is at most 2k − 2 unless N or its dual is free, in which case |E(M) − E(N)| ⩽ k − 1. Examples are given to show that these bounds are best possible for all choices for N. A consequence of the main result is that a minimally connected matroid of rank r and maximum circuit size c has at most 2r − c + 2 elements. This bound sharpens a result of Murty.