3,196 research outputs found
Drawing Trees with Perfect Angular Resolution and Polynomial Area
We study methods for drawing trees with perfect angular resolution, i.e.,
with angles at each node v equal to 2{\pi}/d(v). We show:
1. Any unordered tree has a crossing-free straight-line drawing with perfect
angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any
crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each
edge is represented by a circular arc) with perfect angular resolution and
polynomial area. Thus, our results explore what is achievable with
straight-line drawings and what more is achievable with Lombardi-style
drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure
Lombardi Drawings of Graphs
We introduce the notion of Lombardi graph drawings, named after the American
abstract artist Mark Lombardi. In these drawings, edges are represented as
circular arcs rather than as line segments or polylines, and the vertices have
perfect angular resolution: the edges are equally spaced around each vertex. We
describe algorithms for finding Lombardi drawings of regular graphs, graphs of
bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International
Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure
Achieving Good Angular Resolution in 3D Arc Diagrams
We study a three-dimensional analogue to the well-known graph visualization
approach known as arc diagrams. We provide several algorithms that achieve good
angular resolution for 3D arc diagrams, even for cases when the arcs must
project to a given 2D straight-line drawing of the input graph. Our methods
make use of various graph coloring algorithms, including an algorithm for a new
coloring problem, which we call localized edge coloring.Comment: 12 pages, 5 figures; to appear at the 21st International Symposium on
Graph Drawing (GD 2013
Complexity Analysis of Balloon Drawing for Rooted Trees
In a balloon drawing of a tree, all the children under the same parent are
placed on the circumference of the circle centered at their parent, and the
radius of the circle centered at each node along any path from the root
reflects the number of descendants associated with the node. Among various
styles of tree drawings reported in the literature, the balloon drawing enjoys
a desirable feature of displaying tree structures in a rather balanced fashion.
For each internal node in a balloon drawing, the ray from the node to each of
its children divides the wedge accommodating the subtree rooted at the child
into two sub-wedges. Depending on whether the two sub-wedge angles are required
to be identical or not, a balloon drawing can further be divided into two
types: even sub-wedge and uneven sub-wedge types. In the most general case, for
any internal node in the tree there are two dimensions of freedom that affect
the quality of a balloon drawing: (1) altering the order in which the children
of the node appear in the drawing, and (2) for the subtree rooted at each child
of the node, flipping the two sub-wedges of the subtree. In this paper, we give
a comprehensive complexity analysis for optimizing balloon drawings of rooted
trees with respect to angular resolution, aspect ratio and standard deviation
of angles under various drawing cases depending on whether the tree is of even
or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns
out that some are NP-complete while others can be solved in polynomial time. We
also derive approximation algorithms for those that are intractable in general
Planar and Poly-Arc Lombardi Drawings
In Lombardi drawings of graphs, edges are represented as circular arcs, and
the edges incident on vertices have perfect angular resolution. However, not
every graph has a Lombardi drawing, and not every planar graph has a planar
Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be
drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi
drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing
and further investigate topics connecting planarity and Lombardi drawings.Comment: Expanded version of paper appearing in the 19th International
Symposium on Graph Drawing (GD 2011). 16 pages, 8 figure
A Universal Slope Set for 1-Bend Planar Drawings
We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1)
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