The structure of equivalent 3-separations in a 3-connected matroid

Let M be a matroid. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. This result was extended by Oxley, Semple, and Whittle, who showed that, when M is 3-connected, there is a corresponding tree decomposition that displays all non-trivial 3-separations of M up to a certain natural equivalence. This equivalence is based on the notion of the full closure fcl(Y) of a set Y in M, which is obtained by beginning with Y and alternately applying the closure operators of M and M∗ until no new elements can be added. Two 3-separations (Y1,Y2) and (Z1,Z2) are equivalent if {fcl(Y1),fcl(Y2)}={fcl(Z1),fcl(Z2)}. The purpose of this paper is to identify all the structures in M that lead to two 3-separations being equivalent and to describe the precise role these structures have in determining this equivalence.