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)\ less than or equal to 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. (C) 1998 Elsevier Science B.V. All rights reserved.