68 research outputs found
Labeling Schemes for Bounded Degree Graphs
We investigate adjacency labeling schemes for graphs of bounded degree
. In particular, we present an optimal (up to an additive
constant) adjacency labeling scheme for bounded degree trees.
The latter scheme is derived from a labeling scheme for bounded degree
outerplanar graphs. Our results complement a similar bound recently obtained
for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new
insights for closing the long standing gap for adjacency in trees [Alstrup and
Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree
planar graphs. Finally, we use combinatorial number systems and present an
improved adjacency labeling schemes for graphs of bounded degree with
Spanning Properties of Theta-Theta Graphs
We study the spanning properties of Theta-Theta graphs. Similar in spirit
with the Yao-Yao graphs, Theta-Theta graphs partition the space around each
vertex into a set of k cones, for some fixed integer k > 1, and select at most
one edge per cone. The difference is in the way edges are selected. Yao-Yao
graphs select an edge of minimum length, whereas Theta-Theta graphs select an
edge of minimum orthogonal projection onto the cone bisector. It has been
established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio
11.67, for k' >= 6. In this paper we establish a first spanning ratio of
for Theta-Theta graphs, for the same values of . We also extend the class of
Theta-Theta spanners with parameter 6k', and establish a spanning ratio of
for k' >= 5. We surmise that these stronger results are mainly due to a
tighter analysis in this paper, rather than Theta-Theta being superior to
Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta
graphs decreases to 4.64 as k' increases to 8. These are the first results on
the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table
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
Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators
In this work we consider balanced Schnyder woods for planar graphs, which are
Schnyder woods where the number of incoming edges of each color at each vertex
is balanced as much as possible. We provide a simple linear-time heuristic
leading to obtain well balanced Schnyder woods in practice. As test
applications we consider two important algorithmic problems: the computation of
Schnyder drawings and of small cycle separators. While not being able to
provide theoretical guarantees, our experimental results (on a wide collection
of planar graphs) suggest that the use of balanced Schnyder woods leads to an
improvement of the quality of the layout of Schnyder drawings, and provides an
efficient tool for computing short and balanced cycle separators.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
Lower bounds on the dilation of plane spanners
(I) We exhibit a set of 23 points in the plane that has dilation at least
, improving the previously best lower bound of for the
worst-case dilation of plane spanners.
(II) For every integer , there exists an -element point set
such that the degree 3 dilation of denoted by in the domain of plane geometric spanners. In the
same domain, we show that for every integer , there exists a an
-element point set such that the degree 4 dilation of denoted by
The
previous best lower bound of holds for any degree.
(III) For every integer , there exists an -element point set
such that the stretch factor of the greedy triangulation of is at least
.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2
table
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
Nearest Neighbor Connectivity in Two-Dimensional Multihop MANETs
A Mobile Ad Hoc Network (MANET) is characterized to be a network with free, cooperative, and dynamic nodes, self-organized in a random topology, without any kind of infrastructure, where the communication between two nodes usually occurs using multihop paths. The number of hops used in the multihop path is an important metric for the design and performance analysis of routing protocols in MANETs. In this paper, we derive the probability distribution of the hop count of a multihop path between a source node and a destination node, fixed at a known distance from each other, and when a fixed number of nodes are uniformly distributed in a region of interest. This distribution is obtained by the Poisson randomization method. To obtain the multihop path, we propose a novel routing model in which the nearest distance routing protocol (NR) is analyzed. Numerical results are obtained to evaluate the performance of the NR
Contact Representations of Graphs in 3D
We study contact representations of graphs in which vertices are represented
by axis-aligned polyhedra in 3D and edges are realized by non-zero area common
boundaries between corresponding polyhedra. We show that for every 3-connected
planar graph, there exists a simultaneous representation of the graph and its
dual with 3D boxes. We give a linear-time algorithm for constructing such a
representation. This result extends the existing primal-dual contact
representations of planar graphs in 2D using circles and triangles. While
contact graphs in 2D directly correspond to planar graphs, we next study
representations of non-planar graphs in 3D. In particular we consider
representations of optimal 1-planar graphs. A graph is 1-planar if there exists
a drawing in the plane where each edge is crossed at most once, and an optimal
n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a
linear-time algorithm for representing optimal 1-planar graphs without
separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph
admits a representation with boxes. Hence, we consider contact representations
with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a
quadratic-time algorithm for representing optimal 1-planar graph with L-shaped
polyhedra
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