650 research outputs found
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
A matroid extension result
Adding elements to matroids can be fraught with difficulty. In the V\'amos
matroid , there are four independent sets and such
that is a -separation while exactly three of
the local connectivities , ,
, and are one, with the fourth being
zero. As is well known, there is no extension of by a non-loop element
such that is a circuit for all . This paper proves that a
matroid can be extended by a fixed element in the guts of a -separation
provided no V\'amos-like structure is present
Laminar Matroids
A laminar family is a collection of subsets of a set such
that, for any two intersecting sets, one is contained in the other. For a
capacity function on , let be \{I:|I\cap A|
\leq c(A)\text{ for all A\in\mathscr{A}}\}. Then is the
collection of independent sets of a (laminar) matroid on . We present a
method of compacting laminar presentations, characterize the class of laminar
matroids by their excluded minors, present a way to construct all laminar
matroids using basic operations, and compare the class of laminar matroids to
other well-known classes of matroids.Comment: 17 page
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 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
Generalized Laminar Matroids
Nested matroids were introduced by Crapo in 1965 and have appeared frequently
in the literature since then. A flat of a matroid is Hamiltonian if it has
a spanning circuit. A matroid is nested if and only if its Hamiltonian
flats form a chain under inclusion; is laminar if and only if, for every
-element independent set , the Hamiltonian flats of containing
form a chain under inclusion. We generalize these notions to define the classes
of -closure-laminar and -laminar matroids. This paper focuses on
structural properties of these classes noting that, while the second class is
always minor-closed, the first is if and only if . The main results
are excluded-minor characterizations for the classes of 2-laminar and
2-closure-laminar matroids.Comment: 12 page
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
Unavoidable parallel minors of regular matroids
This is the post-print version of the Article - Copyright @ 2011 ElsevierWe prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M (K_{3,k}), M(W_k), M(K_k), the cycle matroid of the graph obtained from K_{2,k} by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency
Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids
Our splitter theorem for internally 4-connected binary matroids studies pairs
of the form (M,N), where N and M are internally 4-connected binary matroids, M
has a proper N-minor, and if M' is an internally 4-connected matroid such that
M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The
analysis in the splitter theorem requires the constraint that |E(M)|>15. In
this article, we complement that analysis by using an exhaustive computer
search to find all such pairs satisfying |E(M)|<16.Comment: Correcting minor error
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