Matroid 3-connectivity and branch width
  We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X and M/X is 3-connected and contains N as a minor. 
Jim Geelen and Stefan H. M. van Zwam
math.CO