17,791 research outputs found
Straight-line Drawability of a Planar Graph Plus an Edge
We investigate straight-line drawings of topological graphs that consist of a
planar graph plus one edge, also called almost-planar graphs. We present a
characterization of such graphs that admit a straight-line drawing. The
characterization enables a linear-time testing algorithm to determine whether
an almost-planar graph admits a straight-line drawing, and a linear-time
drawing algorithm that constructs such a drawing, if it exists. We also show
that some almost-planar graphs require exponential area for a straight-line
drawing
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
Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling
We present a bijection between some quadrangular dissections of an hexagon
and unrooted binary trees, with interesting consequences for enumeration, mesh
compression and graph sampling. Our bijection yields an efficient uniform
random sampler for 3-connected planar graphs, which turns out to be determinant
for the quadratic complexity of the current best known uniform random sampler
for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also
provides an encoding for the set of -edge 3-connected
planar graphs that matches the entropy bound
bits per edge (bpe). This solves a
theoretical problem recently raised in mesh compression, as these graphs
abstract the combinatorial part of meshes with spherical topology. We also
achieve the {optimal parametric rate} bpe
for graphs of with vertices and faces, matching in
particular the optimal rate for triangulations. Our encoding relies on a linear
time algorithm to compute an orientation associated to the minimal Schnyder
wood of a 3-connected planar map. This algorithm is of independent interest,
and it is for instance a key ingredient in a recent straight line drawing
algorithm for 3-connected planar graphs [\bf Bonichon et al., Graph Drawing
2005]
Planar L-Drawings of Directed Graphs
We study planar drawings of directed graphs in the L-drawing standard. We
provide necessary conditions for the existence of these drawings and show that
testing for the existence of a planar L-drawing is an NP-complete problem.
Motivated by this result, we focus on upward-planar L-drawings. We show that
directed st-graphs admitting an upward- (resp. upward-rightward-) planar
L-drawing are exactly those admitting a bitonic (resp. monotonically
increasing) st-ordering. We give a linear-time algorithm that computes a
bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or
reports that there exists none.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
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
On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
We study two variants of the well-known orthogonal drawing model: (i) the
smooth orthogonal, and (ii) the octilinear. Both models form an extension of
the orthogonal, by supporting one additional type of edge segments (circular
arcs and diagonal segments, respectively).
For planar graphs of max-degree 4, we analyze relationships between the graph
classes that can be drawn bendless in the two models and we also prove
NP-hardness for a restricted version of the bendless drawing problem for both
models. For planar graphs of higher degree, we present an algorithm that
produces bi-monotone smooth orthogonal drawings with at most two segments per
edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Level-Planar Drawings with Few Slopes
We introduce and study level-planar straight-line drawings with a fixed number of slopes. For proper level graphs (all edges connect vertices of adjacent levels), we give an ( log / log log )-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present ( log )-time and ( log )-time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with slopes is NP-hard even in restricted cases
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