1,123,264 research outputs found
Generalised Thurston-Bennequin invariants for real algebraic surface singularities
A generalised Thurston-Bennequin invariant for a Q-singularity of a real
algebraic variety is defined as a linking form on the homologies of the real
link of the singularity. The main goal of this paper is to present a method to
calculate the linking form in terms of the very good resolution graph of a real
normal unibranch surface singularity. For such singularities, the value of the
linking form is the Thurston-Bennequin number of the real link of the
singularity. As a special case of unibranch surface singularities, the
behaviour of the linking form is investigated on the Brieskorn double points
x^m+y^n\pm z^2=0.Comment: 22 pages, TeX, 12 figure
IST Austria Technical Report
We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural properties that arise in program analysis.
We consider that each component of the concurrent system is a graph with constant treewidth, and it is known that the controlflow graphs of most programs have constant treewidth. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis problems (e.g., alias analysis). The study of multiple queries allows us to consider the tradeoff between the resource usage of the \emph{one-time} preprocessing and for \emph{each individual} query. The traditional approaches construct the product graph of all components and apply the best-known graph algorithm on the product. In the traditional approach, even the answer to a single query requires the transitive closure computation (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time.
Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time,
each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results that show that the worst-case running times of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (such as improving
the worst-case bounds for the shortest path problem in general graphs whose current best-known bound has not been improved in five decades). Finally, we provide a prototype implementation of our algorithms which significantly outperforms the existing algorithmic methods on several benchmarks
On graphs of defect at most 2
In this paper we consider the degree/diameter problem, namely, given natural
numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of
vertices in a graph of maximum degree {\Delta} and diameter D. In this context,
the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D).
Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called
Moore graphs, turned out to be very rare. Therefore, it is very interesting to
investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and
order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is,
({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect.
Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1,
({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper
focuses on graphs of defect 2. Building on the approaches developed in [11] we
obtain several new important results on this family of graphs. First, we prove
that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is
2D. Second, and most important, we prove the non-existence of
({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome,
together with a proof on the non-existence of (4, 3,-2)-graphs (also provided
in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs
with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second
census of ({\Delta},D,-2)-graphs known at present, the first being the one of
(3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other
results of this paper include necessary conditions for the existence of
({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the
non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5
such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure
On bipartite graphs of defect at most 4
We consider the bipartite version of the degree/diameter problem, namely,
given natural numbers {\Delta} \geq 2 and D \geq 2, find the maximum number
Nb({\Delta},D) of vertices in a bipartite graph of maximum degree {\Delta} and
diameter D. In this context, the Moore bipartite bound Mb({\Delta},D)
represents an upper bound for Nb({\Delta},D). Bipartite graphs of maximum
degree {\Delta}, diameter D and order Mb({\Delta},D), called Moore bipartite
graphs, have turned out to be very rare. Therefore, it is very interesting to
investigate bipartite graphs of maximum degree {\Delta} \geq 2, diameter D \geq
2 and order Mb({\Delta},D) - \epsilon with small \epsilon > 0, that is,
bipartite ({\Delta},D,-\epsilon)-graphs. The parameter \epsilon is called the
defect. This paper considers bipartite graphs of defect at most 4, and presents
all the known such graphs. Bipartite graphs of defect 2 have been studied in
the past; if {\Delta} \geq 3 and D \geq 3, they may only exist for D = 3.
However, when \epsilon > 2 bipartite ({\Delta},D,-\epsilon)-graphs represent a
wide unexplored area. The main results of the paper include several necessary
conditions for the existence of bipartite -graphs; the complete
catalogue of bipartite (3,D,-\epsilon)-graphs with D \geq 2 and 0 \leq \epsilon
\leq 4; the complete catalogue of bipartite ({\Delta},D,-\epsilon)-graphs with
{\Delta} \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq \epsilon \leq 4; and a
non-existence proof of all bipartite ({\Delta},D,-4)-graphs with {\Delta} \geq
3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs
of defect 4 for {\Delta} \geq 3 and D \geq 5, and comment on some implications
of our results for upper bounds of Nb({\Delta},D).Comment: 25 pages, 14 Postscript figure
Constructions of Large Graphs on Surfaces
We consider the degree/diameter problem for graphs embedded in a surface,
namely, given a surface and integers and , determine the
maximum order of a graph embeddable in with
maximum degree and diameter . We introduce a number of
constructions which produce many new largest known planar and toroidal graphs.
We record all these graphs in the available tables of largest known graphs.
Given a surface of Euler genus and an odd diameter , the
current best asymptotic lower bound for is given by
Our constructions produce
new graphs of order \begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if
$\Sigma$ is the Klein bottle}\\
\(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}&
\text{otherwise,}\end{cases} thus improving the former value by a factor of
4.Comment: 15 pages, 7 figure
- …