A chain theorem for matroids

  • Authors:

  • James Oxley;Charles Semple;Geoff Whittle

  • Affiliations:

  • Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA;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
  • Year:
  • 2008

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Abstract

Tutte's Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)|-|E(N)|=1. Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4-connected, that is, for every 3-separation (X,Y) of N, the set E(N) can be obtained from X or Y by successively applying the operations of closure and coclosure. Hall proved a chain theorem for a different class of 4-connected matroids, those for which every 3-separation has at most five elements on one side. This paper proves a chain theorem for those sequentially 4-connected matroids that also obey this size condition.