612 research outputs found
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions
A greedily routable region (GRR) is a closed subset of , in
which each destination point can be reached from each starting point by
choosing the direction with maximum reduction of the distance to the
destination in each point of the path.
Recently, Tan and Kermarrec proposed a geographic routing protocol for dense
wireless sensor networks based on decomposing the network area into a small
number of interior-disjoint GRRs. They showed that minimum decomposition is
NP-hard for polygons with holes.
We consider minimum GRR decomposition for plane straight-line drawings of
graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing
style which has become a popular research topic in graph drawing. We show that
minimum decomposition is still NP-hard for graphs with cycles, but can be
solved optimally for trees in polynomial time. Additionally, we give a
2-approximation for simple polygons, if a given triangulation has to be
respected.Comment: full version of a paper appearing in ISAAC 201
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
Drawing Graphs as Spanners
We study the problem of embedding graphs in the plane as good geometric
spanners. That is, for a graph , the goal is to construct a straight-line
drawing of in the plane such that, for any two vertices and
of , the ratio between the minimum length of any path from to
and the Euclidean distance between and is small. The maximum such
ratio, over all pairs of vertices of , is the spanning ratio of .
First, we show that deciding whether a graph admits a straight-line drawing
with spanning ratio , a proper straight-line drawing with spanning ratio
, and a planar straight-line drawing with spanning ratio are
NP-complete, -complete, and linear-time solvable problems,
respectively, where a drawing is proper if no two vertices overlap and no edge
overlaps a vertex.
Second, we show that moving from spanning ratio to spanning ratio
allows us to draw every graph. Namely, we prove that, for every
, every (planar) graph admits a proper (resp. planar) straight-line
drawing with spanning ratio smaller than .
Third, our drawings with spanning ratio smaller than have large
edge-length ratio, that is, the ratio between the length of the longest edge
and the length of the shortest edge is exponential. We show that this is
sometimes unavoidable. More generally, we identify having bounded toughness as
the criterion that distinguishes graphs that admit straight-line drawings with
constant spanning ratio and polynomial edge-length ratio from graphs that
require exponential edge-length ratio in any straight-line drawing with
constant spanning ratio
On the Area Requirements of Planar Greedy Drawings of Triconnected Planar Graphs
In this paper we study the area requirements of planar greedy drawings of
triconnected planar graphs. Cao, Strelzoff, and Sun exhibited a family
of subdivisions of triconnected plane graphs and claimed that every planar
greedy drawing of the graphs in respecting the prescribed plane
embedding requires exponential area. However, we show that every -vertex
graph in actually has a planar greedy drawing respecting the
prescribed plane embedding on an grid. This reopens the
question whether triconnected planar graphs admit planar greedy drawings on a
polynomial-size grid. Further, we provide evidence for a positive answer to the
above question by proving that every -vertex Halin graph admits a planar
greedy drawing on an grid. Both such results are obtained by
actually constructing drawings that are convex and angle-monotone. Finally, we
consider -Schnyder drawings, which are angle-monotone and hence greedy
if , and show that there exist planar triangulations for
which every -Schnyder drawing with a fixed requires
exponential area for any resolution rule
On Planar Greedy Drawings of 3-Connected Planar Graphs
A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed.
In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra.
In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra\u27s result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture
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