Constructive characterizations of 3-connected matroids of path width three

A matroid M is sequential or has path width 3 if M is 3-connected and its ground set 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}. This paper proves that every sequential matroid is easily constructible from a uniform matroid of rank or corank two by a sequence of moves each of which consists of a slight modification of segment-cosegment or cosegment-segment exchange. It is also proved that if N is an n-element sequential matroid, then N is representable over all fields with at least n - 1 elements; and there is an attractive family of self-dual sequential 3-connected matroids such that N is a minor of some member of this family. © 2007 Elsevier Ltd. All rights reserved