663 research outputs found
On an excluded-minor class of matroids
A result of Walton and the author establishes that every 3-connected matroid of rank and corank at least three has one of five 6-element rank-3 self-dual matroids as a minor. One of these matroids is the rank-3 whirl W3. Another is the rank-3 matroid P6 that consists of a single 3-point line together with three points off the line. This paper determines the structure of the class of matroids that is obtained by excluding as minors both W3 and P6. As a consequence of this result, we deduce a characterization of the class of GF(4)-representable matroids with no W3-minor. © 1990
Computing excluded minors for classes of matroids representable over partial fields
We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids
There are only a finite number of excluded minors for the class of bicircular matroids
We show that the class of bicircular matroids has only a finite number of
excluded minors. Key tools used in our proof include representations of
matroids by biased graphs and the recently introduced class of quasi-graphic
matroids. We show that if is an excluded minor of rank at least ten, then
is quasi-graphic. Several small excluded minors are quasi-graphic. Using
biased-graphic representations, we find that already contains one of these.
We also provide an upper bound, in terms of rank, on the number of elements in
an excluded minor, so the result follows.Comment: Added an appendix describing all known excluded minors. Added Gordon
Royle as author. Some proofs revised and correcte
Excluded minors for the class of split matroids
The class of split matroids arises by putting conditions on the system of
split hyperplanes of the matroid base polytope. It can alternatively be defined
in terms of structural properties of the matroid. We use this structural
description to give an excluded minor characterisation of the class
Defining bicircular matroids in monadic logic
We conjecture that the class of frame matroids can be characterised by a
sentence in the monadic second-order logic of matroids, and we prove that there
is such a characterisation for the class of bicircular matroids. The proof does
not depend on an excluded-minor characterisation
Structural Results for Matroids.
This dissertation solves some problems involving the structure of matroids. In Chapter 2, the class of binary matroids with no minors isomorphic to the prism graph, its dual, and the binary affine cube is completely determined. This class contains the infinite family of matroids obtained by sticking together a wheel and the Fano matroid across a triangle, and then deleting an edge of the triangle. In Chapter 3, we extend a graph result by D. W. Hall to matroids. Hall proved that if a simple, 3-connected graph has a K\sb5-minor, then it must also have a K\sb{3,3}-minor, the only exception being K\sb5 itself. We prove that if a 3-connected, binary matroid has an M(K\sb5)-minor, then it must also have a minor isomorphic to M(K\sb{3,3}) or its dual, the only exceptions being M(K\sb5), a highly symmetric 12-element matroid T\sb{12}, and T\sb{12} with any edge contracted. Chapter 4 consists of a collection of results on the intersection of circuits and cocircuits in binary matroids. In Chapter 5, we describe, in terms of excluded minors, the class of non-binary matroids with the property that a matroid is in the class if its restriction to every hyperplane is binary
Matroids Whose Ground Sets are Domains of Functions
From an integer-valued function f we obtain, in a natural way, a matroid Mf on the domain of f. We show that the class of matroids so obtained is closed under restriction, contraction, duality, truncation and elongation, but not under direct sum. We give an excluded-minor characterization of and show that consists precisely of those transversal matroids with a presentation in which the sets in the presentation are nested. Finally, we show that on an n-set there are exactly 2”. © 1982, Australian Mathematical Society. All rights reserved
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