945 research outputs found
Obstructions to within a few vertices or edges of acyclic
Finite obstruction sets for lower ideals in the minor order are guaranteed to
exist by the Graph Minor Theorem. It has been known for several years that, in
principle, obstruction sets can be mechanically computed for most natural lower
ideals. In this paper, we describe a general-purpose method for finding
obstructions by using a bounded treewidth (or pathwidth) search. We illustrate
this approach by characterizing certain families of cycle-cover graphs based on
the two well-known problems: -{\sc Feedback Vertex Set} and -{\sc
Feedback Edge Set}. Our search is based on a number of algorithmic strategies
by which large constants can be mitigated, including a randomized strategy for
obtaining proofs of minimality.Comment: 16 page
From Frazier-Jawerth characterizations of Besov spaces to Wavelets and Decomposition spaces
This article describes how the ideas promoted by the fundamental papers
published by M. Frazier and B. Jawerth in the eighties have influenced
subsequent developments related to the theory of atomic decompositions and
Banach frames for function spaces such as the modulation spaces and
Besov-Triebel-Lizorkin spaces.
Both of these classes of spaces arise as special cases of two different,
general constructions of function spaces: coorbit spaces and decomposition
spaces. Coorbit spaces are defined by imposing certain decay conditions on the
so-called voice transform of the function/distribution under consideration. As
a concrete example, one might think of the wavelet transform, leading to the
theory of Besov-Triebel-Lizorkin spaces.
Decomposition spaces, on the other hand, are defined using certain
decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one
uses a dyadic decomposition, while a uniform decomposition yields modulation
spaces.
Only recently, the second author has established a fruitful connection
between modern variants of wavelet theory with respect to general dilation
groups (which can be treated in the context of coorbit theory) and a particular
family of decomposition spaces. In this way, optimal inclusion results and
invariance properties for a variety of smoothness spaces can be established. We
will present an outline of these connections and comment on the basic results
arising in this context
Definability Equals Recognizability for -Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph
property that can be defined by a sentence in counting monadic second order
logic (CMSOL) can be checked in linear time for graphs of bounded treewidth,
which is known as Courcelle's Theorem. These algorithms are constructed as
finite state tree automata, and hence every CMSOL-definable graph property is
recognizable. Courcelle also conjectured that the converse holds, i.e. every
recognizable graph property is definable in CMSOL for graphs of bounded
treewidth. We prove this conjecture for -outerplanar graphs, which are known
to have treewidth at most .Comment: 40 pages, 8 figure
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
In the companion paper [Linear rank-width of distance-hereditary graphs I. A
polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a
characterization of the linear rank-width of distance-hereditary graphs, from
which we derived an algorithm to compute it in polynomial time. In this paper,
we investigate structural properties of distance-hereditary graphs based on
this characterization.
First, we prove that for a fixed tree , every distance-hereditary graph of
sufficiently large linear rank-width contains a vertex-minor isomorphic to .
We extend this property to bigger graph classes, namely, classes of graphs
whose prime induced subgraphs have bounded linear rank-width. Here, prime
graphs are graphs containing no splits. We conjecture that for every tree ,
every graph of sufficiently large linear rank-width contains a vertex-minor
isomorphic to . Our result implies that it is sufficient to prove this
conjecture for prime graphs.
For a class of graphs closed under taking vertex-minors, a graph
is called a vertex-minor obstruction for if but all of
its proper vertex-minors are contained in . Secondly, we provide, for
each , a set of distance-hereditary graphs that contains all
distance-hereditary vertex-minor obstructions for graphs of linear rank-width
at most . Also, we give a simpler way to obtain the known vertex-minor
obstructions for graphs of linear rank-width at most .Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1
Sparse Matrix Decompositions and Graph Characterizations
The question of when zeros (i.e., sparsity) in a positive definite matrix
are preserved in its Cholesky decomposition, and vice versa, was addressed by
Paulsen et al. in the Journal of Functional Analysis (85, pp151-178). In
particular, they prove that for the pattern of zeros in to be retained in
the Cholesky decomposition of , the pattern of zeros in has to
necessarily correspond to a chordal (or decomposable) graph associated with a
specific type of vertex ordering. This result therefore yields a
characterization of chordal graphs in terms of sparse positive definite
matrices. It has also proved to be extremely useful in probabilistic and
statistical analysis of Markov random fields where zeros in positive definite
correlation matrices are intimately related to the notion of stochastic
independence. Now, consider a positive definite matrix and its Cholesky
decomposition given by , where is lower triangular with unit
diagonal entries, and a diagonal matrix with positive entries. In this
paper, we prove that a necessary and sufficient condition for zeros (i.e.,
sparsity) in a positive definite matrix to be preserved in its associated
Cholesky matrix , \, and in addition also preserved in the inverse of the
Cholesky matrix , is that the pattern of zeros corresponds to a
co-chordal or homogeneous graph associated with a specific type of vertex
ordering. We proceed to provide a second characterization of this class of
graphs in terms of determinants of submatrices that correspond to cliques in
the graph. These results add to the growing body of literature in the field of
sparse matrix decompositions, and also prove to be critical ingredients in the
probabilistic analysis of an important class of Markov random fields
Different forms of metric characterizations of classes of Banach spaces
For each sequence X of finite-dimensional Banach spaces there exists a
sequence H of finite connected nweighted graphs with maximum degree 3 such that
the following conditions on a Banach space Y are equivalent: (1) Y admits
uniformly isomorphic embeddings of elements of the sequence X. (2) Y admits
uniformly bilipschitz embeddings of elements of the sequence H.Comment: Accepted for publication in Houston Journal of Mathematic
Characterizing Sparse Graphs by Map Decompositions
A map is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into k edge-disjoint maps after: (1) the addition of any â„“ edges; (2) the addition of some â„“ edges. These graphs are identified with classes of sparse graphs; the results are also given in matroidal terms
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