236,429 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
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
Drawing Graphs within Restricted Area
We study the problem of selecting a maximum-weight subgraph of a given graph
such that the subgraph can be drawn within a prescribed drawing area subject to
given non-uniform vertex sizes. We develop and analyze heuristics both for the
general (undirected) case and for the use case of (directed) calculation graphs
which are used to analyze the typical mistakes that high school students make
when transforming mathematical expressions in the process of calculating, for
example, sums of fractions
Monotone Grid Drawings of Planar Graphs
A monotone drawing of a planar graph is a planar straight-line drawing of
where a monotone path exists between every pair of vertices of in some
direction. Recently monotone drawings of planar graphs have been proposed as a
new standard for visualizing graphs. A monotone drawing of a planar graph is a
monotone grid drawing if every vertex in the drawing is drawn on a grid point.
In this paper we study monotone grid drawings of planar graphs in a variable
embedding setting. We show that every connected planar graph of vertices
has a monotone grid drawing on a grid of size , and such a
drawing can be found in O(n) time
Signed graph embedding: when everybody can sit closer to friends than enemies
Signed graphs are graphs with signed edges. They are commonly used to
represent positive and negative relationships in social networks. While balance
theory and clusterizable graphs deal with signed graphs to represent social
interactions, recent empirical studies have proved that they fail to reflect
some current practices in real social networks. In this paper we address the
issue of drawing signed graphs and capturing such social interactions. We relax
the previous assumptions to define a drawing as a model in which every vertex
has to be placed closer to its neighbors connected via a positive edge than its
neighbors connected via a negative edge in the resulting space. Based on this
definition, we address the problem of deciding whether a given signed graph has
a drawing in a given -dimensional Euclidean space. We present forbidden
patterns for signed graphs that admit the introduced definition of drawing in
the Euclidean plane and line. We then focus on the -dimensional case, where
we provide a polynomial time algorithm that decides if a given complete signed
graph has a drawing, and constructs it when applicable
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
Drawing Big Graphs using Spectral Sparsification
Spectral sparsification is a general technique developed by Spielman et al.
to reduce the number of edges in a graph while retaining its structural
properties. We investigate the use of spectral sparsification to produce good
visual representations of big graphs. We evaluate spectral sparsification
approaches on real-world and synthetic graphs. We show that spectral
sparsifiers are more effective than random edge sampling. Our results lead to
guidelines for using spectral sparsification in big graph visualization.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
- …