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

Abstract

Tutte defined a k-separation of a matroid M to be a partition ðA; BÞ of the ground set of M such that jAj; jBjXk and rðAÞ þ rðBÞ À rðMÞok: If, for all mon; the matroid M has no mseparations, then M is n-connected. Earlier, Whitney showed that ðA; BÞ 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: r

An Algorithm for Constructing a k-Tree for a k-Connected Matroid

“…The requirement that fcl k (P 1 ∪ P 2 ∪ · · · ∪ P j ) = E(M ), not present in the k = 3 case, is necessary, as will become evident in Example 5.3. Corollary 3.10 generalises the corresponding results for k = 3 [10,Corollary 5.10] and k = 4 [2,Corollary 3.15]. Corollary 3.11 is a straightforward generalisation of [9,Corollary 3.5] that follows from Corollaries 3.9 and 3.10.…”

“…The fcl k operator is a well-defined closure operator on the set of exactly k-separating subsets of E [4,Lemma 3.3]. When k = 3, the operator is equivalent to the full closure operator for 3-connected matroids (as given in [10], for example) and, when k = 4, it is equivalent to the full 2-span operator of [2]. It is important to note that the full k-closure operator is only well-defined on exactly k-separating sets; that is, k-separating sets with at least k − 1 elements, but no more than |E| − (k − 1) elements.…”

“…Informally, one can obtain a paddle by gluing together sufficiently structured matroids along a common line, called the spine. For further details, see [10,Section 4]. The free (n, j)-swirl is a 3-connected matroid obtained by beginning with a basis {1, 2, .…”

“…Clark and Whittle [4] generalised the main result of [10], showing that every tangle of order k in a connectivity system that satisfies a certain "robustness" property has a tree decomposition, called a k-tree, that displays, up to equivalence, all the non-sequential k-separations of the connectivity system with respect to the tangle. In particular, this result specialises to k-connected matroids as follows: Theorem 1.1.…”

“…Oxley et al [10] showed that every 3-connected matroid M with at least nine elements has a 3-tree: a tree decomposition that displays, up to a natural equivalence, all non-sequential 3-separations of M . The approach taken in the proof of this result does not appear to elicit an efficient algorithm for finding such a 3-tree.…”