11 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
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
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
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 figure
Angles of Arc-Polygons and Lombardi Drawings of Cacti
We characterize the triples of interior angles that are possible in
non-self-crossing triangles with circular-arc sides, and we prove that a given
cyclic sequence of angles can be realized by a non-self-crossing polygon with
circular-arc sides whenever all angles are at most pi. As a consequence of
these results, we prove that every cactus has a planar Lombardi drawing (a
drawing with edges depicted as circular arcs, meeting at equal angles at each
vertex) for its natural embedding in which every cycle of the cactus is a face
of the drawing. However, there exist planar embeddings of cacti that do not
have planar Lombardi drawings.Comment: 12 pages, 8 figures. To be published in Proc. 33rd Canadian
Conference on Computational Geometry, 202
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
The Graphs of Planar Soap Bubbles
We characterize the graphs formed by two-dimensional soap bubbles as being
exactly the 3-regular bridgeless planar multigraphs. Our characterization
combines a local characterization of soap bubble graphs in terms of the
curvatures of arcs meeting at common vertices, a proof that this
characterization remains invariant under Moebius transformations, an
application of Moebius invariance to prove bridgelessness, and a
Moebius-invariant power diagram of circles previously developed by the author
for its applications in graph drawing.Comment: 16 pages, 9 figure