19 research outputs found
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Drawing graphs with vertices and edges in convex position
A graph has strong convex dimension , if it admits a straight-line drawing
in the plane such that its vertices are in convex position and the midpoints of
its edges are also in convex position. Halman, Onn, and Rothblum conjectured
that graphs of strong convex dimension are planar and therefore have at
most edges. We prove that all such graphs have at most edges
while on the other hand we present a class of non-planar graphs of strong
convex dimension . We also give lower bounds on the maximum number of edges
a graph of strong convex dimension can have and discuss variants of this
graph class. We apply our results to questions about large convexly independent
sets in Minkowski sums of planar point sets, that have been of interest in
recent years.Comment: 15 pages, 12 figures, improved expositio
Obstacle Numbers of Planar Graphs
Given finitely many connected polygonal obstacles in the
plane and a set of points in general position and not in any obstacle, the
{\em visibility graph} of with obstacles is the (geometric)
graph with vertex set , where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph
is the smallest integer such that is the visibility graph of a set of
points with obstacles. If is planar, we define the planar obstacle
number of by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of ). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order is
, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least is .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Maxent-Stress Optimization of 3D Biomolecular Models
Knowing a biomolecule\u27s structure is inherently linked to and a prerequisite for any detailed understanding of its function. Significant effort has gone into developing technologies for structural characterization. These technologies do not directly provide 3D structures; instead they typically yield noisy and erroneous distance information between specific entities such as atoms or residues, which have to be translated into consistent 3D models.
Here we present an approach for this translation process based on maxent-stress optimization. Our new approach extends the original graph drawing method for the new application\u27s specifics by introducing additional constraints and confidence values as well as algorithmic components. Extensive experiments demonstrate that our approach infers structural models (i.e., sensible 3D coordinates for the molecule\u27s atoms) that correspond well to the distance information, can handle noisy and error-prone data, and is considerably faster than established tools. Our results promise to allow domain scientists nearly-interactive structural modeling based on distance constraints
Generating realistic scaled complex networks
Research on generative models is a central project in the emerging field of
network science, and it studies how statistical patterns found in real networks
could be generated by formal rules. Output from these generative models is then
the basis for designing and evaluating computational methods on networks, and
for verification and simulation studies. During the last two decades, a variety
of models has been proposed with an ultimate goal of achieving comprehensive
realism for the generated networks. In this study, we (a) introduce a new
generator, termed ReCoN; (b) explore how ReCoN and some existing models can be
fitted to an original network to produce a structurally similar replica, (c)
use ReCoN to produce networks much larger than the original exemplar, and
finally (d) discuss open problems and promising research directions. In a
comparative experimental study, we find that ReCoN is often superior to many
other state-of-the-art network generation methods. We argue that ReCoN is a
scalable and effective tool for modeling a given network while preserving
important properties at both micro- and macroscopic scales, and for scaling the
exemplar data by orders of magnitude in size.Comment: 26 pages, 13 figures, extended version, a preliminary version of the
paper was presented at the 5th International Workshop on Complex Networks and
their Application
Algorithms and Bounds for Drawing Directed Graphs
In this paper we present a new approach to visualize directed graphs and
their hierarchies that completely departs from the classical four-phase
framework of Sugiyama and computes readable hierarchical visualizations that
contain the complete reachability information of a graph. Additionally, our
approach has the advantage that only the necessary edges are drawn in the
drawing, thus reducing the visual complexity of the resulting drawing.
Furthermore, most problems involved in our framework require only polynomial
time. Our framework offers a suite of solutions depending upon the
requirements, and it consists of only two steps: (a) the cycle removal step (if
the graph contains cycles) and (b) the channel decomposition and hierarchical
drawing step. Our framework does not introduce any dummy vertices and it keeps
the vertices of a channel vertically aligned. The time complexity of the main
drawing algorithms of our framework is , where is the number of
channels, typically much smaller than (the number of vertices).Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
The Widths of Strict Outerconfluent Graphs
Strict outerconfluent drawing is a style of graph drawing in which vertices
are drawn on the boundary of a disk, adjacencies are indicated by the existence
of smooth curves through a system of tracks within the disk, and no two
adjacent vertices are connected by more than one of these smooth tracks. We
investigate graph width parameters on the graphs that have drawings in this
style. We prove that the clique-width of these graphs is unbounded, but their
twin-width is bounded.Comment: 15 pages, 2 figure