4,013 research outputs found
Restricted String Representations
A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph
has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called -VPG representations) and simultaneously two curves intersect each other at most once (those are the
so-called 1-string representations). Thus, planar graphs are -VPG -string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The -VPG representations of planar graphs satisfy these restrictions. We attempt to further
restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate -VPG
representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation
of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string
Planar graphs as L-intersection or L-contact graphs
The L-intersection graphs are the graphs that have a representation as
intersection graphs of axis parallel shapes in the plane. A subfamily of these
graphs are {L, |, --}-contact graphs which are the contact graphs of axis
parallel L, |, and -- shapes in the plane. We prove here two results that were
conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are
L-intersection graphs, and that triangle-free planar graphs are {L, |,
--}-contact graphs. These results are obtained by a new and simple
decomposition technique for 4-connected triangulations. Our results also
provide a much simpler proof of the known fact that planar graphs are segment
intersection graphs
On Semantic Word Cloud Representation
We study the problem of computing semantic-preserving word clouds in which
semantically related words are close to each other. While several heuristic
approaches have been described in the literature, we formalize the underlying
geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this
model each word is associated with rectangle with fixed dimensions, and the
goal is to represent semantically related words by ensuring that the two
corresponding rectangles touch. We design and analyze efficient polynomial-time
algorithms for some variants of the WRAC problem, show that several general
variants are NP-hard, and describe a number of approximation algorithms.
Finally, we experimentally demonstrate that our theoretically-sound algorithms
outperform the early heuristics
Connectivity Compression for Irregular Quadrilateral Meshes
Applications that require Internet access to remote 3D datasets are often
limited by the storage costs of 3D models. Several compression methods are
available to address these limits for objects represented by triangle meshes.
Many CAD and VRML models, however, are represented as quadrilateral meshes or
mixed triangle/quadrilateral meshes, and these models may also require
compression. We present an algorithm for encoding the connectivity of such
quadrilateral meshes, and we demonstrate that by preserving and exploiting the
original quad structure, our approach achieves encodings 30 - 80% smaller than
an approach based on randomly splitting quads into triangles. We present both a
code with a proven worst-case cost of 3 bits per vertex (or 2.75 bits per
vertex for meshes without valence-two vertices) and entropy-coding results for
typical meshes ranging from 0.3 to 0.9 bits per vertex, depending on the
regularity of the mesh. Our method may be implemented by a rule for a
particular splitting of quads into triangles and by using the compression and
decompression algorithms introduced in [Rossignac99] and
[Rossignac&Szymczak99]. We also present extensions to the algorithm to compress
meshes with holes and handles and meshes containing triangles and other
polygons as well as quads
On grounded L-graphs and their relatives
We consider the graph class Grounded-L corresponding to graphs that admit an
intersection representation by L-shaped curves, where additionally the topmost
points of each curve are assumed to belong to a common horizontal line. We
prove that Grounded-L graphs admit an equivalent characterisation in terms of
vertex ordering with forbidden patterns.
We also compare this class to related intersection classes, such as the
grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance
graphs), or the outer-1-string graphs. We give constructions showing that these
classes are all distinct and satisfy only trivial or previously known
inclusions.Comment: 16 pages, 6 figure
Wilson loops in 3-dimensional N=6 supersymmetric Chern-Simons Theory and their string theory duals
We study Wilson loops in the three-dimensional N=6 supersymmetric
Chern-Simons theory recently constructed by Aharony, Bergman, Jafferis and
Maldacena, that is conjectured to be dual to type IIA string theory on AdS_4 x
CP^3. We construct loop operators in the Chern-Simons theory which preserve 1/6
of the supercharges and calculate their expectation value up to 2-loop order at
weak coupling. The expectation value at strong coupling is found by
constructing the string theory duals of these operators. For low dimensional
representations these are fundamental strings, for high dimensional
representations these are D2-branes and D6-branes. In support of this
identification we demonstrate that these string theory solutions match the
symmetries, charges and the preserved supersymmetries of their Chern-Simons
theory counterparts.Comment: 28 pages. v2: references added, choice of the Wilson loop operator
clarified; v3: combinatorial factor of 2 in perturbative calculation
corrected; v4: typos corrected, version to be publishe
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