Obstructions to branch-decomposition of matroids

A (delta,gamma)-net in a matroid M is a pair (N, P) where N is a minor of M, P is a set of series classes in N, vertical bar P vertical bar >=delta, and the pairwise connectivity, in M, between any two members of P is at least gamma. We prove that, for any finite field F, nets provide a qualitative characterization for branch-width in the class of F-representable matroids. That is, for an F-representable matroid M, we prove that: (1) if M contains a (delta, gamma)-net where delta and gamma are both very large, then M has large branch-width, and, conversely, (2) if the branch-width of M is very large, then M or M* contains a (delta, gamma)-net where delta and gamma are both large. (C) 2005 Elsevier Inc. All rights reserved.