1,322 research outputs found
The Determination of a Matroid\u27s Structure From Properties of Certain Large Minors.
This dissertation solves some problems related to the structure of matroids. In Chapter 2, we prove that if M and N are distinct connected matroids on a common ground set E, where and, for every e in or M/e = N/e, then one of M and N is a relaxation of the other. In addition, we determine the matroids M and N on a common ground set E such that, for every pair of elements of E, at least two of the four corresponding minors of M and N obtained by eliminating e and f are equal. The theorems in Chapter 3 and 4 extend a result of Oxley that characterizes the non-binary matroids M such that, for each element e, or M/e is binary. In Chapter 3, we describe the non-binary matroids M such that, for every pair of elements at least two of the four minors of M obtained by eliminating e and f are binary. In Chapter 4, we obtain an alternative extension of Oxley\u27s result by changing the minor under consideration from the smallest 3-connected whirl, U\sb{2,4}, to the smallest 3-connected wheel, M(K\sb4). In particular, we determine the binary matroids M having an M(K\sb4)-minor such that, for every element e, or M/e has no M(K\sb4)-minor. This enables us to characterize the matroids M that are not series-parallel networks, but, for every or M/e is a series-parallel network
Matroids with at least two regular elements
For a matroid , an element such that both and
are regular is called a regular element of . We determine completely the
structure of non-regular matroids with at least two regular elements. Besides
four small size matroids, all 3-connected matroids in the class can be pieced
together from or and a regular matroid using 3-sums. This result
takes a step toward solving a problem posed by Paul Seymour: Find all
3-connected non-regular matroids with at least one regular element [5, 14.8.8]
A notion of minor-based matroid connectivity
For a matroid , a matroid is -connected if every two elements of
are in an -minor together. Thus a matroid is connected if and only if it
is -connected. This paper proves that is the only connected
matroid such that if is -connected with , then or is -connected for all elements . Moreover, we
show that and are the only connected matroids
such that, whenever a matroid has an -minor using and an -minor
using , it also has an -minor using . Finally, we show
that is -connected if and only if every clonal
class of is trivial.Comment: 13 page
On two classes of nearly binary matroids
We give an excluded-minor characterization for the class of matroids M in
which M\e or M/e is binary for all e in E(M). This class is closely related to
the class of matroids in which every member is binary or can be obtained from a
binary matroid by relaxing a circuit-hyperplane. We also provide an
excluded-minor characterization for the second class.Comment: 14 pages, 4 figures. This paper has been accepted for publication in
the European Journal of Combinatorics. This is the final version of the pape
Complementation, Local Complementation, and Switching in Binary Matroids
In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex
set can be obtained from a complete graph on via a sequence of the
operations of complementation, switching edges and non-edges at a vertex, and
local complementation. The last operation involves taking the complement in the
neighbourhood of a vertex. In this paper, we consider natural generalizations
of these operations for binary matroids and explore their behaviour. We
characterize all binary matroids obtainable from the binary projective geometry
of rank under the operations of complementation and switching. Moreover, we
show that not all binary matroids of rank at most can be obtained from a
projective geometry of rank via a sequence of the three generalized
operations. We introduce a fourth operation and show that, with this additional
operation, we are able to obtain all binary matroids.Comment: Fixed an error in the proof of Theorem 5.3. Adv. in Appl. Math.
(2020
A decomposition theorem for binary matroids with no prism minor
The prism graph is the dual of the complete graph on five vertices with an
edge deleted, . In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in . Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, and ,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of , the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5]
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
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