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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. © 2005 Elsevier Inc. All rights reserved.

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Advances in Applied Mathematics

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