16 research outputs found
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
Pixel and Voxel Representations of Graphs
We study contact representations for graphs, which we call pixel
representations in 2D and voxel representations in 3D. Our representations are
based on the unit square grid whose cells we call pixels in 2D and voxels in
3D. Two pixels are adjacent if they share an edge, two voxels if they share a
face. We call a connected set of pixels or voxels a blob. Given a graph, we
represent its vertices by disjoint blobs such that two blobs contain adjacent
pixels or voxels if and only if the corresponding vertices are adjacent. We are
interested in the size of a representation, which is the number of pixels or
voxels it consists of.
We first show that finding minimum-size representations is NP-complete. Then,
we bound representation sizes needed for certain graph classes. In 2D, we show
that, for -outerplanar graphs with vertices, pixels are
always sufficient and sometimes necessary. In particular, outerplanar graphs
can be represented with a linear number of pixels, whereas general planar
graphs sometimes need a quadratic number. In 3D, voxels are
always sufficient and sometimes necessary for any -vertex graph. We improve
this bound to for graphs of treewidth and to
for graphs of genus . In particular, planar graphs
admit representations with voxels
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
Small-Area Orthogonal Drawings of 3-Connected Graphs
It is well-known that every graph with maximum degree 4 has an orthogonal drawing with area at most (49/64)*n^2+O(n)â0.76n^2. In this paper, we show that if the graph is 3-connected, then the area can be reduced even further to (9/16)*n^2+O(n)â0.56n^2. The drawing uses the 3-canonical order for (not necessarily planar) 3-connected graphs, which is a special Mondshein sequence and can hence be computed in linear time. To our knowledge, this is the first application of a Mondshein sequence in graph drawing
How to morph a tree on a small grid
In this paper we study planar morphs between straight-line planar grid
drawings of trees. A morph consists of a sequence of morphing steps, where in a
morphing step vertices move along straight-line trajectories at constant speed.
We show how to construct planar morphs that simultaneously achieve a reduced
number of morphing steps and a polynomially-bounded resolution. We assume that
both the initial and final drawings lie on the grid and we ensure that each
morphing step produces a grid drawing; further, we consider both upward
drawings of rooted trees and drawings of arbitrary trees.Comment: A preliminary version of this paper appears in WADS 201
1-Bend Upward Planar Drawings of SP-Digraphs
It is proved that every series-parallel digraph whose maximum vertex-degree is ÎÎ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Î distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with optimal angular resolution Ï/Î . A variant of the proof technique is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) one-bend planar drawing with âÎ/2â distinct slopes if biconnected, and with âÎ/2â+1 distinct slopes if connected
Degree Aware Triangulation of Annular Regions
Generating constrained triangulation of point sites distributed in the plane is an important problem in computational geometry. We present theoretical and experimental investigation results for generating triangulations for polygons and point sites that address node degree constraints. We characterize point sites that have almost all vertices of odd degree. We present experimental results on the node degree distribution of Delaunay triangulation of point sites generated randomly. Additionally, we present a heuristic algorithm for triangulating a given normal annular region with increased number of even degree vertices