31 research outputs found
Splitters and Decomposers for Binary Matroids
Let denote the class of binary matroids with no minors
isomorphic to . In this paper we give a decomposition theorem
for , where is a certain 10-element rank-4
matroid. As corollaries we obtain decomposition theorems for the classes
obtained by excluding the Kuratowski graphs and . These decomposition
theorems imply results on internally -connected matroids by Zhou
[\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte
[\ref{Mayhewsubmitted}].Comment: arXiv admin note: text overlap with arXiv:1403.775
On Binary And Regular Matroids Without Small Minors
The results of this dissertation consist of excluded-minor results for Binary Matroids and excluded-minor results for Regular Matroids. Structural theorems on the relationship between minors and k-sums of matroids are developed here in order to provide some of these characterizations. Chapter 2 of the dissertation contains excluded-minor results for Binary Matroids. The first main result of this dissertation is a characterization of the internally 4-connected binary matroids with no minor that is isomorphic to the cycle matroid of the prism+e graph. This characterization generalizes results of Mayhew and Royle [18] for binary matroids and results of Dirac [8] and Lovász [15] for graphs. The results of this chapter are then extended from the class of internally 4-connected matroids to the class of 3-connected matroids. Chapter 3 of the dissertation contains the second main result, a decomposition theorem for regular matroids without certain minors. This decomposition theorem is used to obtain excluded-minor results for Regular Matroids. Wagner, Lovász, Oxley, Ding, Liu, and others have characterized many classes of graphs that are H-free for graphs H with at most twelve edges (see [7]). We extend several of these excluded-minor characterizations to regular matroids in Chapter 3. We also provide characterizations of regular matroids excluding several graphic matroids such as the octahedron, cube, and the Möbius Ladder on eight vertices. Both theoretical and computer-aided proofs of the results of Chapters 2 and 3 are provided in this dissertation
Beta invariant and variations of chain theorems for matroids
The beta invariant of a matroid was introduced by Crapo in 1967. We first find the lower bound of the beta invariant of 3-connected matroids with rank r and the matroids which attain the lower bound. Second we characterize the matroids with beta invariant 5 and 6. For binary matroids we characterize matroids with beta invariant 7. These results extend earlier work of Oxley. Lastly we partially answer an open question of chromatic uniqueness of wheels and prove a splitting formula for the beta invariant of generalized parallel connection of two matroids. Tutte\u27s Wheel-and-Whirl theorem and Seymour\u27s Splitter theorem give respectively a constructive and structural view of the 3-connected matroids. Geelen and Whittle proved a chain theorem for sequentially 4-connected matroids and Geelen and Zhou proved a chain theorem for weakly 4-connected matroids. From these theorems one can obtain a chain theorem for matroids as well. We prove a chain theorem for sequentially 4-connected and weakly 4-connected matroids
The Contributions of Dominic Welsh to Matroid Theory
Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh\u27s work in and influence on the development of matroid theory
Nowhere-zero 4-flow in almost Petersen-minor free graphs
AbstractTutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15–20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let (P10)μ̄ be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless (P10)3̄-minor free graph admits a nowhere-zero 4-flow
Aspects of matroid connectivity and uniformity.
In approaching a combinatorial problem, it is often desirable to be armed with
a notion asserting that some objects are more highly structured than others. In
particular, focusing on highly structured objects may avoid certain degeneracies
and allow for the core of the problem to be addressed. In matroid theory, the
principle notion fulfilling this role of “structure” is that of connectivity. This
thesis proves a number of results furthering the knowledge of matroid connectivity
and also introduces a new structural notion, that of generalised uniformity.
The first part of this thesis considers 3-connected matroids and the presence
of elements which may be deleted or contracted without the introduction of any
non-minimal 2-separations. Principally, a Wheels-and-Whirls Theorem and then
a Splitter Theorem is established, guaranteeing the existence of such elements,
provided certain well-behaved structures are not present.
The second part of this thesis generalises the notion of a uniform matroid
by way of a 2-parameter property capturing “how uniform” a given matroid is.
Initially, attention is focused on matroids representable over some field. In particular,
a finiteness result is established and a specific class of binary matroids is
completely determined. The concept of generalised uniformity is then considered
more broadly by an analysis of its relevance to a number of established matroid
notions and settings. Within that analysis, a number of equivalent characterisations
of generalised uniformity are obtained.
Lastly, the third part of the thesis considers a highly structured class of
matroids whose members are defined by the nature of their circuits. A characterisation
is achieved for the regular members of this class and, in general, the
infinitely many excluded series minors are determined
The search for an excluded minor characterization of ternary Rayleigh matroids
Rayleigh matroids are a class of matroids with sets of bases that satisfy
a strong negative correlation property. Interesting characteristics include
the existence of an efficient algorithm for sampling the bases of a Rayleigh
matroid [7]. It has been conjectured that the class of Rayleigh matroids
satisfies Mason’s conjecture [14]. Though many elementary properties of
Rayleigh matroids have been established, it is not known if this class has a
finite set of minimal excluded minors. At this time, it seems unlikely that this
is the case. It has been shown that there is a single minimal excluded minor
for the smaller class of binary Rayleigh matroids [5]. The aim of this thesis
is to detail our search for the set of minimal excluded minors for ternary
Rayleigh matroids. We have found several minimal excluded minors for the
above class of matroids. However, our search is incomplete. It is unclear
whether the set of excluded minors for this set of matroids is finite or not,
and, if finite, what the complete set of minimal excluded minors is. For
our method to answer this question definitively will require a new computer
program. This program would automate a step in our process that we have
done by hand: writing polynomials in at least ten indeterminates as a sum
with many terms, squared
Packing odd -joins with at most two terminals
Take a graph , an edge subset , and a set of
terminals where is even. The triple is
called a signed graft. A -join is odd if it contains an odd number of edges
from . Let be the maximum number of edge-disjoint odd -joins.
A signature is a set of the form where and is even. Let be the minimum cardinality a -cut
or a signature can achieve. Then and we say that
packs if equality holds here.
We prove that packs if the signed graft is Eulerian and it
excludes two special non-packing minors. Our result confirms the Cycling
Conjecture for the class of clutters of odd -joins with at most two
terminals. Corollaries of this result include, the characterizations of weakly
and evenly bipartite graphs, packing two-commodity paths, packing -joins
with at most four terminals, and a new result on covering edges with cuts.Comment: extended abstract appeared in IPCO 2014 (under the different title
"the cycling property for the clutter of odd st-walks"