The structure of 3-connected matroids of path width three

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 (e1, e2, ..., en) such that ({e1, e2, ..., ek}, {ek + 1, ek + 2, ..., en}) 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. © 2005 Elsevier Ltd. All rights reserved