313 research outputs found
Tangle-tree duality: in graphs, matroids and beyond
We apply a recent duality theorem for tangles in abstract separation systems
to derive tangle-type duality theorems for width-parameters in graphs and
matroids. We further derive a duality theorem for the existence of clusters in
large data sets.
Our applications to graphs include new, tangle-type, duality theorems for
tree-width, path-width, and tree-decompositions of small adhesion. Conversely,
we show that carving width is dual to edge-tangles. For matroids we obtain a
duality theorem for tree-width.
Our results can be used to derive short proofs of all the classical duality
theorems for width parameters in graph minor theory, such as path-width,
tree-width, branch-width and rank-width.Comment: arXiv admin note: text overlap with arXiv:1406.379
The structure of 2-separations of infinite matroids
Generalizing a well known theorem for finite matroids, we prove that for
every (infinite) connected matroid M there is a unique tree T such that the
nodes of T correspond to minors of M that are either 3-connected or circuits or
cocircuits, and the edges of T correspond to certain nested 2-separations of M.
These decompositions are invariant under duality.Comment: 31 page
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
Separations of Matroid Freeness Properties
Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest
An obstacle to a decomposition theorem for near-regular matroids
Seymour's Decomposition Theorem for regular matroids states that any matroid
representable over both GF(2) and GF(3) can be obtained from matroids that are
graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped
that similar characterizations hold for other classes of matroids, notably for
the class of near-regular matroids. Suppose that all near-regular matroids can
be obtained from matroids that belong to a few basic classes through k-sums.
Also suppose that these basic classes are such that, whenever a class contains
all graphic matroids, it does not contain all cographic matroids. We show that
in that case 3-sums will not suffice.Comment: 11 pages, 1 figur
A chain theorem for 4-connected matroids
For the abstract of this paper, please see the PDF file
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