3,608 research outputs found
Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs
In this paper, we revisit the split decomposition of graphs and give new
combinatorial and algorithmic results for the class of totally decomposable
graphs, also known as the distance hereditary graphs, and for two non-trivial
subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give
strutural and incremental characterizations, leading to optimal fully-dynamic
recognition algorithms for vertex and edge modifications, for each of these
classes. These results rely on a new framework to represent the split
decomposition, namely the graph-labelled trees, which also captures the modular
decomposition of graphs and thereby unify these two decompositions techniques.
The point of the paper is to use bijections between these graph classes and
trees whose nodes are labelled by cliques and stars. Doing so, we are also able
to derive an intersection model for distance hereditary graphs, which answers
an open problem.Comment: extended abstract appeared in ISAAC 2007: Dynamic distance hereditary
graphs using split decompositon. In International Symposium on Algorithms and
Computation - ISAAC. Number 4835 in Lecture Notes, pages 41-51, 200
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
The fully residually F quotients of F*<x,y>
We describe the fully residually F; or limit groups relative to F; (where F
is a free group) that arise from systems of equations in two variables over F
that have coefficients in F.Comment: 64 pages, 2 figures. Following recommendations from a referee, the
paper has been completely reorganized and many small mistakes have been
corrected. There were also a few gaps in the earlier version of the paper
that have been fixed. In particular much of the content of Section 8 in the
previous version had to be replaced. This paper is to appear in Groups. Geom.
Dy
Bayesian clustering in decomposable graphs
In this paper we propose a class of prior distributions on decomposable
graphs, allowing for improved modeling flexibility. While existing methods
solely penalize the number of edges, the proposed work empowers practitioners
to control clustering, level of separation, and other features of the graph.
Emphasis is placed on a particular prior distribution which derives its
motivation from the class of product partition models; the properties of this
prior relative to existing priors is examined through theory and simulation. We
then demonstrate the use of graphical models in the field of agriculture,
showing how the proposed prior distribution alleviates the inflexibility of
previous approaches in properly modeling the interactions between the yield of
different crop varieties.Comment: 3 figures, 1 tabl
Polar syzygies in characteristic zero: the monomial case
Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field
of characteristic zero, we focus on the first syzygy module Z of the transposed
Jacobian module D(f), whose elements are called differential syzygies of f.
There is a distinct submodule P of Z coming from the polynomial relations of f
through its transposed Jacobian matrix, the elements of which are called polar
syzygies of f. We say that f is polarizable if equality P=Z holds. This paper
is concerned with the situation where f are monomials of degree 2, in which
case one can naturally associate to them a graph G(f) with loops and translate
the problem into a combinatorial one. A main result is a complete combinatorial
characterization of polarizability in terms of special configurations in this
graph. As a consequence, we show that polarizability implies normality of the
subalgebra k[f] of R and that the converse holds provided the graph G(f) is
free of certain degenerate configurations. One main combinatorial class of
polarizability is the class of polymatroidal sets. We also prove that if the
edge graph of G(f) has diameter at most 2 then f is polarizable. We establish a
curious connection with birationality of rational maps defined by monomial
quadrics.Comment: 33 pages, 15 figure
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