On the 3-connected matroids that are minimal having a fixed spanning restriction

Let N be a minor of a 3-connected matroid M and let M′ be a 3-connected minor of M that is minimal having N as a minor. This paper commences the study of the problem of finding a best-possible upper bound on |E(M′)−E(N)| . The main result solves this problem in the case that N and M have the same rank.