4,759 research outputs found
Empty triangles in good drawings of the complete graph
A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph Kn with n vertices is at least n.Peer ReviewedPostprint (author’s final draft
Convex drawings of the complete graph: topology meets geometry
In this work, we introduce and develop a theory of convex drawings of the
complete graph in the sphere. A drawing of is convex if, for
every 3-cycle of , there is a closed disc bounded by
such that, for any two vertices with and both in
, the entire edge is also contained in .
As one application of this perspective, we consider drawings containing a
non-convex that has restrictions on its extensions to drawings of .
For each such drawing, we use convexity to produce a new drawing with fewer
crossings. This is the first example of local considerations providing
sufficient conditions for suboptimality. In particular, we do not compare the
number of crossings {with the number of crossings in} any known drawings. This
result sheds light on Aichholzer's computer proof (personal communication)
showing that, for , every optimal drawing of is convex.
Convex drawings are characterized by excluding two of the five drawings of
. Two refinements of convex drawings are h-convex and f-convex drawings.
The latter have been shown by Aichholzer et al (Deciding monotonicity of good
drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational
Geometry (EGC 2015), 2015) and, independently, the authors of the current
article (Levi's Lemma, pseudolinear drawings of , and empty triangles,
\rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear
drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as
demonstrated recently by Arroyo et al (Extending drawings of complete graphs
into arrangements of pseudocircles, submitted)
Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition
A geometric graph is angle-monotone if every pair of vertices has a path
between them that---after some rotation---is - and -monotone.
Angle-monotone graphs are -spanners and they are increasing-chord
graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in
2014 and proved that Gabriel triangulations are angle-monotone graphs. We give
a polynomial time algorithm to recognize angle-monotone geometric graphs. We
prove that every point set has a plane geometric graph that is generalized
angle-monotone---specifically, we prove that the half--graph is
generalized angle-monotone. We give a local routing algorithm for Gabriel
triangulations that finds a path from any vertex to any vertex whose
length is within times the Euclidean distance from to .
Finally, we prove some lower bounds and limits on local routing algorithms on
Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends
We study the following classes of beyond-planar graphs: 1-planar, IC-planar,
and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar,
and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every
edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs
of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two
pairs of crossing edges share two vertices. We study the relations of these
beyond-planar graph classes (beyond-planar graphs is a collective term for the
primary attempts to generalize the planar graphs) to right-angle crossing (RAC)
graphs that admit compact drawings on the grid with few bends. We present four
drawing algorithms that preserve the given embeddings. First, we show that
every -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . Finally, we make two
known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at
most one bend per edge and for drawing IC-planar RAC graphs straight-line
Witness Gabriel Graphs
We consider a generalization of the Gabriel graph, the witness Gabriel graph.
Given a set of vertices P and a set of witnesses W in the plane, there is an
edge ab between two points of P in the witness Gabriel graph GG-(P,W) if and
only if the closed disk with diameter ab does not contain any witness point
(besides possibly a and/or b). We study several properties of the witness
Gabriel graph, both as a proximity graph and as a new tool in graph drawing.Comment: 23 pages. EuroCG 200
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