The structure of 3-connected matroids of path width three

  • Authors:

  • Rhiannon Hall;James Oxley;Charles Semple

  • Affiliations:

  • School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, UB8 3PH, United Kingdom;Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA;Department of Mathematics and Statistics, University of Canterbury, Christchurch 8020, New Zealand

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

A 3-connected matroid M is sequential or has path width 3 if its ground set E(M) has a sequential ordering, that is, an ordering (e"1,e"2,...,e"n) such that ({e"1,e"2,...,e"k},{e"k"+"1,e"k"+"2,...,e"n}) is a 3-separation for all k in {3,4,...,n-3}. In this paper, we consider the possible sequential orderings that such a matroid can have. In particular, we prove that M essentially has two fixed ends, each of which is a maximal segment, a maximal cosegment, or a maximal fan. We also identify the possible structures in M that account for different sequential orderings of E(M). These results rely on an earlier paper of the authors that describes the structure of equivalent non-sequential 3-separations in a 3-connected matroid. Those results are extended here to describe the structure of equivalent sequential 3-separations.