27 research outputs found
The -connected Excluded Minors for the Class of Quasi-graphic Matroids
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards,
and Whittle, is minor closed and contains both the class of lifted-graphic
matroids and the class of frame matroids, each of which generalises the class
of graphic matroids. In this paper, we prove that the matroids and
are the only -connected excluded minors for the class of
quasi-graphic matroids
Templates for Representable Matroids
The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors.
Chapter 1 gives a brief introduction to matroids and matroid structure theory. Chapters 2 and 3 analyze this hypothesis of Geelen, Gerards, and Whittle and propose some refined hypotheses. In Chapter 3, we define frame templates and discuss various notions of template equivalence.
Chapter 4 gives some details on how templates relate to each other. We define a preorder on the set of frame templates over a finite field, and we determine the minimal nontrivial templates with respect to this preorder. We also study in significant depth a specific type of template that is pertinent to many applications. Chapters 5 and 6 apply the results of Chapters 3 and 4 to several subclasses of the binary matroids and the quaternary matroids---those matroids representable over the fields of two and four elements, respectively.
Two of the classes we study in Chapter 5 are the even-cycle matroids and the even-cut matroids. Each of these classes has hundreds of excluded minors. We show that, for highly connected matroids, two or three excluded minors suffice. We also show that Seymour\u27s 1-Flowing Conjecture holds for sufficiently highly connected matroids.
In Chapter 6, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the extremal functions for these classes, verifying a conjecture of Archer for matroids of sufficiently large rank
Describing Quasi-Graphic Matroids
The class of quasi-graphic matroids recently introduced by Geelen, Gerards,
and Whittle generalises each of the classes of frame matroids and
lifted-graphic matroids introduced earlier by Zaslavsky. For each biased graph
Zaslavsky defined a unique lift matroid
and a unique frame matroid , each on ground set . We
show that in general there may be many quasi-graphic matroids on and
describe them all. We provide cryptomorphic descriptions in terms of subgraphs
corresponding to circuits, cocircuits, independent sets, and bases. Equipped
with these descriptions, we prove some results about quasi-graphic matroids. In
particular, we provide alternate proofs that do not require 3-connectivity of
two results of Geelen, Gerards, and Whittle for 3-connected matroids from their
introductory paper: namely, that every quasi-graphic matroid linearly
representable over a field is either lifted-graphic or frame, and that if a
matroid has a framework with a loop that is not a loop of then is
either lifted-graphic or frame. We also provide sufficient conditions for a
quasi-graphic matroid to have a unique framework.
Zaslavsky has asked for those matroids whose independent sets are contained
in the collection of independent sets of while containing
those of , for some biased graph . Adding a
natural (and necessary) non-degeneracy condition defines a class of matroids,
which we call biased graphic. We show that the class of biased graphic matroids
almost coincides with the class of quasi-graphic matroids: every quasi-graphic
matroid is biased graphic, and if is a biased graphic matroid that is not
quasi-graphic then is a 2-sum of a frame matroid with one or more
lifted-graphic matroids
The Templates for Some Classes of Quaternary Matroids
Subject to hypotheses based on the matroid structure theory of Geelen,
Gerards, and Whittle, we completely characterize the highly connected members
of the class of golden-mean matroids and several other closely related classes
of quaternary matroids. This leads to a determination of the eventual extremal
functions for these classes. One of the main tools for obtaining these results
is the notion of a frame template. Consequently, we also study frame templates
in significant depth.Comment: 83 pages; minor corrections in Version 4; accepted for publication by
Journal of Combinatorial Theory, Series
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
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
Representations of even-cycle and even-cut matroids
In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair where is a graph and is a subset of edges of .
A cycle of is a subset of edges of such that every vertex of the subgraph of induced by has an even degree. We say that is even in if is even. A matroid is an even-cycle matroid if there exists a signed graph such that circuits of precisely corresponds to inclusion-wise minimal non-empty even cycles of . A graft is a pair where is a graph and is a subset of vertices of such that each component of contains an even number of vertices in . Let be a subset of vertices of and let be a cut of . We say that is even in if is even. A matroid is an even-cut matroid if there exists a graft such that circuits of corresponds to inclusion-wise minimal non-empty even cuts of .\\
This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations.
(a) Isomorphism problem: what is the relationship between two representations?
(b) Bounding the number of representations: how many representations can a matroid have?
(c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists?
These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two -connected graphs and have a pair of signatures such that and represent the same even-cycle matroids. This also characterize when and have a pair of terminal sets such that and represent the same even-cut matroid.
For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is -connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by . An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a -matrix over the finite field \gf(2). The time-complexity of these algorithms is polynomial in the size of the input matrix