The structure of the 3-separations of 3-connected matroids

  • Authors:

  • James Oxley;Charles Semple;Geoff Whittle

  • Affiliations:

  • Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana;Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand;School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand

  • Venue:
  • Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
  • Year:
  • 2004

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Abstract

Tutte defined a k-separation of a matroid M to be a partition (A, B) of the ground set of M such that |A|, |B| ≥ k and r(A) + r(B) - r(M) k. If, for all m n, the matroid M has no m- separations, then M is n-connected. Earlier, Whitney showed that (A, B) is a 1-separation of M if and only if A is a union of 2-connected components of M. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2- separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M.