141 research outputs found
Excluding Kuratowski graphs and their duals from binary matroids
We consider some applications of our characterisation of the internally
4-connected binary matroids with no M(K3,3)-minor. We characterise the
internally 4-connected binary matroids with no minor in some subset of
{M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We
also describe a practical algorithm for testing whether a binary matroid has a
minor in the subset. In addition we characterise the growth-rate of binary
matroids with no M(K3,3)-minor, and we show that a binary matroid with no
M(K3,3)-minor has critical exponent over GF(2) at most equal to four.Comment: Some small change
Fixing numbers for matroids
Motivated by work in graph theory, we define the fixing number for a matroid.
We give upper and lower bounds for fixing numbers for a general matroid in
terms of the size and maximum orbit size (under the action of the matroid
automorphism group). We prove the fixing numbers for the cycle matroid and
bicircular matroid associated with 3-connected graphs are identical. Many of
these results have interpretations through permutation groups, and we make this
connection explicit.Comment: This is a major revision of a previous versio
Infinite Matroids and Determinacy of Games
Solving a problem of Diestel and Pott, we construct a large class of infinite
matroids. These can be used to provide counterexamples against the natural
extension of the Well-quasi-ordering-Conjecture to infinite matroids and to
show that the class of planar infinite matroids does not have a universal
matroid.
The existence of these matroids has a connection to Set Theory in that it
corresponds to the Determinacy of certain games. To show that our construction
gives matroids, we introduce a new very simple axiomatization of the class of
countable tame matroids
Constructing internally 4-connected binary matroids
This is the post-print version of the Article - Copyright @ 2013 ElsevierIn an earlier paper, we proved that an internally 4-connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in that it produces a smaller minor from the original. We also describe each as a bottom-up operation, constructing a larger matroid from the original, and we give necessary and su fficient conditions for each of these bottom-up moves to produce an internally 4-connected binary matroid. From this, we derive a constructive method for generating all internally 4-connected binary matroids.This study is supported by NSF IRFP Grant 0967050, the Marsden Fund, and the National Security Agency
Towards a splitter theorem for internally 4-connected binary matroids
This is the post-print version of the Article - Copyright @ 2012 ElsevierWe prove that if M is a 4-connected binary matroid and N is an internally 4-connected proper minor of M with at least 7 elements, then, unless M is a certain 16-element matroid, there is an element e of E(M) such that either M\e or M/e is internally 4-connected having an N-minor. This strengthens a result of Zhou and is a first step towards obtaining a splitter theorem for internally 4-connected binary matroids.This study is partially funded by Marsden Fund of New Zealand and the National Security Agency
COMs: Complexes of Oriented Matroids
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured
them as asymmetric counterparts of oriented matroids, both sharing the key
property of strong elimination. Moreover, symmetry of faces holds in both
structures as well as in the so-called affine oriented matroids. These two
fundamental properties (formulated for covectors) together lead to the natural
notion of "conditional oriented matroid" (abbreviated COM). These novel
structures can be characterized in terms of three cocircuits axioms,
generalizing the familiar characterization for oriented matroids. We describe a
binary composition scheme by which every COM can successively be erected as a
certain complex of oriented matroids, in essentially the same way as a lopsided
set can be glued together from its maximal hypercube faces. A realizable COM is
represented by a hyperplane arrangement restricted to an open convex set. Among
these are the examples formed by linear extensions of ordered sets,
generalizing the oriented matroids corresponding to the permutohedra. Relaxing
realizability to local realizability, we capture a wider class of combinatorial
objects: we show that non-positively curved Coxeter zonotopal complexes give
rise to locally realizable COMs.Comment: 40 pages, 6 figures, (improved exposition
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