The structure of 3-connected matroids of path width three

A 3-connected matroid MM is sequential or has path width 3 if its ground set E(M)E(M) has a sequential ordering, that is, an ordering (e1,e2,…,en)(e1,e2,…,en) such that ({e1,e2,…,ek},{ek+1,ek+2,…,en})({e1,e2,…,ek},{ek+1,ek+2,…,en}) is a 3-separation for all kk in {3,4,…,n−3}{3,4,…,n−3}. In this paper, we consider the possible sequential orderings that such a matroid can have. In particular, we prove that MM 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 MM that account for different sequential orderings of E(M)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.