755 research outputs found
Computing ?-Stretch Paths in Drawings of Graphs
Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing ?=f(P), is ?-stretch if ? is a simple (non-self-intersecting) curve, and for every pair of distinct points p?P and q?P, the length of the sub-curve of ? connecting f(p) with f(q) is at most ?||f(p)-f(q)?, where ?.? denotes the Euclidean distance. We introduce and study the ?-stretch Path Problem (?SP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a ?-stretch path P connecting s and t. The ?SP also asks that we output P if it exists.
The ?SP quantifies a notion of "near straightness" for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a ?-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes.
We prove that ?SP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ?>0, ??O(poly(log |V(G)|)), and s,t?V(G), outputs a ?-stretch path between s and t, if a (1-?)?-stretch path between s and t exists in the drawing
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
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
Drawing Trees with Perfect Angular Resolution and Polynomial Area
We study methods for drawing trees with perfect angular resolution, i.e.,
with angles at each node v equal to 2{\pi}/d(v). We show:
1. Any unordered tree has a crossing-free straight-line drawing with perfect
angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any
crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each
edge is represented by a circular arc) with perfect angular resolution and
polynomial area. Thus, our results explore what is achievable with
straight-line drawings and what more is achievable with Lombardi-style
drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure
Aligned Drawings of Planar Graphs
Let be a graph that is topologically embedded in the plane and let
be an arrangement of pseudolines intersecting the drawing of .
An aligned drawing of and is a planar polyline drawing
of with an arrangement of lines so that and are
homeomorphic to and . We show that if is
stretchable and every edge either entirely lies on a pseudoline or it has
at most one intersection with , then and have a
straight-line aligned drawing. In order to prove this result, we strengthen a
result of Da Lozzo et al., and prove that a planar graph and a single
pseudoline have an aligned drawing with a prescribed convex
drawing of the outer face. We also study the less restrictive version of the
alignment problem with respect to one line, where only a set of vertices is
given and we need to determine whether they can be collinear. We show that the
problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th
International Symposium on Graph Drawing and Network Visualization (GD 2017
The Partial Visibility Representation Extension Problem
For a graph , a function is called a \emph{bar visibility
representation} of when for each vertex , is a
horizontal line segment (\emph{bar}) and iff there is an
unobstructed, vertical, -wide line of sight between and
. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
, a bar visibility representation of , additionally, puts the bar
strictly below the bar for each directed edge of
. We study a generalization of the recognition problem where a function
defined on a subset of is given and the question is whether
there is a bar visibility representation of with for every . We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
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