The Structure of Equivalent 3-separations in a 3-connected Matroid
نویسنده
چکیده
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.
منابع مشابه
The Structure of the 4-separations in 4-connected Matroids
For a 2-connected matroid M , Cunningham and Edmonds gave a tree decomposition that displays all of its 2-separations. When M is 3-connected, two 3-separations are equivalent if one can be obtained from the other by passing through a sequence of 3-separations each of which is obtained from its predecessor by moving a single element from one side of the 3-separation to the other. Oxley, Semple, ...
متن کاملThe structure of 3-connected matroids of path width three
A 3-connected matroid M is sequential or has path width 3 if its ground set E(M) has a sequential ordering, that is, an ordering (e1, e2, . . . , en) such that ({e1, e2, . . . , ek}, {ek+1, ek+2, . . . , en}) is a 3-separation for all k in {3, 4, . . . , n − 3}. In this paper, we consider the possible sequential orderings that such a matroid can have. In particular, we prove that M essentially ...
متن کاملExposing 3-separations in 3-connected matroids
Let M be a 3-connected matroid other than a wheel or a whirl. In the next paper in this series, we prove that there is an element whose deletion from M or M∗ is 3-connected and whose only 3separations are equivalent to those induced by M . The strategy used to prove this theorem involves showing that we can remove some element from a leaf of the tree of 3-separations of M . The main result of t...
متن کاملThe Structure of Crossing Separations in Matroids
Oxley, Semple and Whittle described a tree decomposition for a 3-connected matroid M that displays, up to a natural equivalence, all non-trivial 3-separations of M . Crossing 3-separations gave rise to fundamental structures known as flowers. In this paper, we define a generalized flower structure called a k-flower, with no assumptions on the connectivity of M . We completely classify k-flowers...
متن کاملThe 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 tre...
متن کامل