6,284 research outputs found
Drawing Graphs with Circular Arcs and Right-Angle Crossings
In a RAC drawing of a graph, vertices are represented by points in the plane,
adjacent vertices are connected by line segments, and crossings must form right
angles. Graphs that admit such drawings are RAC graphs. RAC graphs are
beyond-planar graphs and have been studied extensively. In particular, it is
known that a RAC graph with n vertices has at most 4n - 10 edges.
We introduce a superclass of RAC graphs, which we call arc-RAC graphs. A
graph is arc-RAC if it admits a drawing where edges are represented by circular
arcs and crossings form right angles. We provide a Tur\'an-type result showing
that an arc-RAC graph with n vertices has at most 14n - 12 edges and that there
are n-vertex arc-RAC graphs with 4.5n - o(n) edges
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
Axis-Parallel Right Angle Crossing Graphs
A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity.
In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model
Axis-Parallel Right Angle Crossing Graphs
A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in
which each crossing occurs at a right angle. Originally motivated by
psychological studies on readability of graph layouts, RAC graphs form one of
the most prominent graph classes in beyond planarity.
In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or
apRAC, for short), that restricts the crossings to pairs of axis-parallel
edge-segments. apRAC drawings combine the readability of planar drawings with
the clarity of (non-planar) orthogonal drawings. We consider these graphs both
with and without bends. Our contribution is as follows: (i) We study inclusion
relationships between apRAC and traditional RAC graphs. (ii) We establish
bounds on the edge density of apRAC graphs. (iii) We show that every graph with
maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some
of our results on apRAC graphs also improve the state of the art for general
RAC graphs. We conclude our work with a list of open questions and a discussion
of a natural generalization of the apRAC model
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
Simultaneous Embeddings with Few Bends and Crossings
A simultaneous embedding with fixed edges (SEFE) of two planar graphs and
is a pair of plane drawings of and that coincide when restricted to
the common vertices and edges of and . We show that whenever and
admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve
with few bends and every pair of edges has few crossings. Specifically: (1) if
and are trees then one bend per edge and four crossings per edge pair
suffice (and one bend per edge is sometimes necessary), (2) if is a planar
graph and is a tree then six bends per edge and eight crossings per edge
pair suffice, and (3) if and are planar graphs then six bends per edge
and sixteen crossings per edge pair suffice. Our results improve on a paper by
Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a
paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge
pair suffice.Comment: Full version of the paper "Simultaneous Embeddings with Few Bends and
Crossings" accepted at GD '1
A Coloring Algorithm for Disambiguating Graph and Map Drawings
Drawings of non-planar graphs always result in edge crossings. When there are
many edges crossing at small angles, it is often difficult to follow these
edges, because of the multiple visual paths resulted from the crossings that
slow down eye movements. In this paper we propose an algorithm that
disambiguates the edges with automatic selection of distinctive colors. Our
proposed algorithm computes a near optimal color assignment of a dual collision
graph, using a novel branch-and-bound procedure applied to a space
decomposition of the color gamut. We give examples demonstrating the
effectiveness of this approach in clarifying drawings of real world graphs and
maps
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