46 research outputs found
Algorithms and Bounds for Drawing Non-planar Graphs with Crossing-free Subgraphs
We initiate the study of the following problem: Given a non-planar graph G
and a planar subgraph S of G, does there exist a straight-line drawing {\Gamma}
of G in the plane such that the edges of S are not crossed in {\Gamma} by any
edge of G? We give positive and negative results for different kinds of
connected spanning subgraphs S of G. Moreover, in order to enlarge the subset
of instances that admit a solution, we consider the possibility of bending the
edges of G not in S; in this setting we discuss different trade-offs between
the number of bends and the required drawing area.Comment: 21 pages, 9 figures, extended version of 'Drawing Non-planar Graphs
with Crossing-free Subgraphs' (21st International Symposium on Graph Drawing,
2013
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
A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings
A plus-contact representation of a planar graph is called -balanced if
for every plus shape , the number of other plus shapes incident to each
arm of is at most , where is the maximum degree
of . Although small values of have been achieved for a few subclasses of
planar graphs (e.g., - and -trees), it is unknown whether -balanced
representations with exist for arbitrary planar graphs.
In this paper we compute -balanced plus-contact representations for
all planar graphs that admit a rectangular dual. Our result implies that any
graph with a rectangular dual has a 1-bend box-orthogonal drawings such that
for each vertex , the box representing is a square of side length
.Comment: A poster related to this research appeared at the 25th International
Symposium on Graph Drawing & Network Visualization (GD 2017
On Compact RAC Drawings
We present new bounds for the required area of Right Angle Crossing (RAC) drawings for complete graphs, i.e. drawings where any two crossing edges are perpendicular to each other. First, we improve upon results by Didimo et al. [Walter Didimo et al., 2011] and Di Giacomo et al. [Emilio Di Giacomo et al., 2011] by showing how to compute a RAC drawing with three bends per edge in cubic area. We also show that quadratic area can be achieved when allowing eight bends per edge in general or with three bends per edge for p-partite graphs. As a counterpart, we prove that in general quadratic area is not sufficient for RAC drawings with three bends per edge
L-Drawings of Directed Graphs
We introduce L-drawings, a novel paradigm for representing directed graphs
aiming at combining the readability features of orthogonal drawings with the
expressive power of matrix representations. In an L-drawing, vertices have
exclusive - and -coordinates and edges consist of two segments, one
exiting the source vertically and one entering the destination horizontally.
We study the problem of computing L-drawings using minimum ink. We prove its
NP-completeness and provide a heuristics based on a polynomial-time algorithm
that adds a vertex to a drawing using the minimum additional ink. We performed
an experimental analysis of the heuristics which confirms its effectiveness.Comment: 11 pages, 7 figure