12 research outputs found
An upgraded Wheels-and-Whirls Theorem for 3-connected matroids
Let M be a 3-connected matroid that is not a wheel or a whirl. In this paper, we prove that M has an element e such that M\e or M/e is 3-connected and has no 3-separation that is not equivalent to one induced by M. © 2011 Elsevier Inc
Matroid 3-connectivity and branch width
We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X and M/X is 3-connected and contains N as a minor
Matroid 3-connectivity and branch width
We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element subset X of E(M) such that one of M\X and M/X is 3-connected and contains N as a minor
Matroid 3-connectivity and branch width
We prove that, for each nonnegative integer k and each matroid N, if M is a
3-connected matroid containing N as a minor, and the the branch width of M is
sufficiently large, then there is a k-element subset X of E(M) such that one of
M\X and M/X is 3-connected and contains N as a minor.Comment: 21 page
Inequivalent Representations of Matroids over Prime Fields
It is proved that for each prime field , there is an integer
such that a 4-connected matroid has at most inequivalent representations
over . We also prove a stronger theorem that obtains the same conclusion
for matroids satisfying a connectivity condition, intermediate between
3-connectivity and 4-connectivity that we term "-coherence".
We obtain a variety of other results on inequivalent representations
including the following curious one. For a prime power , let denote the set of matroids representable over all fields with at least
elements. Then there are infinitely many Mersenne primes if and only if,
for each prime power , there is an integer such that a 3-connected
member of has at most inequivalent
GF(7)-representations.
The theorems on inequivalent representations of matroids are consequences of
structural results that do not rely on representability. The bulk of this paper
is devoted to proving such results
Aspects of Matroid Connectivity
Connectivity is a fundamental tool for matroid theorists, which has become increasingly important in the eventual solution of many problems in matroid theory. Loosely speaking, connectivity can be used to help describe a matroid's structure. In this thesis, we prove a series of results that further the knowledge and understanding in the field of matroid connectivity. These results fall into two parts.
First, we focus on 3-connected matroids. A chain theorem is a result that proves the existence of an element, or elements, whose deletion or contraction preserves a predetermined connectivity property. We prove a series of chain theorems for 3-connected matroids where, after fixing a basis B, the elements in B are only eligible for contraction, while the elements not in B are only eligible for deletion. Moreover, we prove a splitter theorem, where a 3-connected minor is also preserved, resolving a conjecture posed by Whittle and Williams in 2013.
Second, we consider k-connected matroids, where k >= 3. A certain tree, known as a k-tree, can be used to describe the structure of a k-connected matroid. We present an algorithm for constructing a k-tree for a k-connected matroid M. Provided that the rank of a subset of E(M) can be found in unit time, the algorithm runs in time polynomial in |E(M)|. This generalises Oxley and Semple's (2013) polynomial-time algorithm for constructing a 3-tree for a 3-connected matroid
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, and Whittle gave a tree decomposition that displays, up to this equivalence, all non-trivial 3-separations of M. Now let M be 4-connected. In this paper, we define two 4-separations of M to be 2-equivalent if one can be obtained from the other by passing through a sequence of 4-separations each obtained from its predecessor by moving at most two elements from one side of the 4-separation to the other. The main result of the paper proves that M has a tree decomposition that displays, up to 2-equivalence, all non-trivial 4-separations of M. © 2011 Elsevier Inc. All rights reserved
Constructing a 3-tree for a 3-connected matroid
In an earlier paper with Whittle, we showed that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid M. The purpose of this paper is to give a polynomial-time algorithm for constructing such a tree for M. © 2012 Elsevier Inc
Exposing 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 3-separations 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 this paper is designed to allow us to do this. © 2010 Elsevier Inc