The structure of the 3-separations of 3-connected matroids

Tutte defined a k–separation of a matroid M to be a partition (A, B) of the ground set of M such that |A|, |B| ≥ k and r(A) + r(B) − r(M) < k. If, for all m < n, the matroid M has no m–separations, then M is n–connected. Earlier, Whitney showed that (A, B) is a 1–separation of M if and only if A is a union of 2–connected components of M . When M is 2–connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2–separations. When M is 3–connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3– separations of M .