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

Abstract

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 ( Y 1 , Y 2 ) and ( Z 1 , Z 2 ) are equivalent if { fcl ( Y 1 ) , fcl ( Y 2 ) } = { fcl ( Z 1 ) , fcl ( Z 2 ) } . 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.