Obstructions to branch-decomposition of matroids

  • Authors:

  • J. Geelen;B. Gerards;N. Robertson;G. Whittle

  • Affiliations:

  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada;CWI, GB Amsterdam, The Netherlands and Department of Mathematics and Computer Science, Eindhoven University of Technology, MB Eindhoven, The Netherlands;Department of Mathematics, Ohio State University, Columbus, OH;School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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Abstract

A (δ, γ)-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, |P| ≥ δ, and the pairwise connectivity, in M, between any two members of P is at least γ. 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 (δ, γ)-net where δ and γ 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 (δ, γ)-net where δ and γ are both large.