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 ( 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.