32 research outputs found
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Monotone Grid Drawings of Planar Graphs
A monotone drawing of a planar graph is a planar straight-line drawing of
where a monotone path exists between every pair of vertices of in some
direction. Recently monotone drawings of planar graphs have been proposed as a
new standard for visualizing graphs. A monotone drawing of a planar graph is a
monotone grid drawing if every vertex in the drawing is drawn on a grid point.
In this paper we study monotone grid drawings of planar graphs in a variable
embedding setting. We show that every connected planar graph of vertices
has a monotone grid drawing on a grid of size , and such a
drawing can be found in O(n) time
A Center Transversal Theorem for Hyperplanes and Applications to Graph Drawing
Motivated by an open problem from graph drawing, we study several
partitioning problems for line and hyperplane arrangements. We prove a
ham-sandwich cut theorem: given two sets of n lines in R^2, there is a line l
such that in both line sets, for both halfplanes delimited by l, there are
n^{1/2} lines which pairwise intersect in that halfplane, and this bound is
tight; a centerpoint theorem: for any set of n lines there is a point such that
for any halfplane containing that point there are (n/3)^{1/2} of the lines
which pairwise intersect in that halfplane. We generalize those results in
higher dimension and obtain a center transversal theorem, a same-type lemma,
and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This
is done by formulating a generalization of the center transversal theorem which
applies to set functions that are much more general than measures. Back to
Graph Drawing (and in the plane), we completely solve the open problem that
motivated our search: there is no set of n labelled lines that are universal
for all n-vertex labelled planar graphs. As a side note, we prove that every
set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar
graphs
Non-Homotopic Drawings of Multigraphs
A multigraph drawn in the plane is non-homotopic if no two edges connecting
the same pair of vertices can be continuously deformed into each other without
passing through a vertex, and is -crossing if every pair of edges
(self-)intersects at most times. We prove that the number of edges in an
-vertex non-homotopic -crossing multigraph is at most ,
which is a big improvement over previous upper bounds.
We also study this problem in the setting of monotone drawings where every
edge is an x-monotone curve. We show that the number of edges, , in such a
drawing is at most and the number of crossings is
. For fixed these bounds
are both best possible up to a constant multiplicative factor.Comment: 19 page
Counting Carambolas
We give upper and lower bounds on the maximum and minimum number of geometric
configurations of various kinds present (as subgraphs) in a triangulation of
points in the plane. Configurations of interest include \emph{convex
polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also
consider related problems for \emph{directed} planar straight-line graphs.Comment: update reflects journal version, to appear in Graphs and
Combinatorics; 18 pages, 13 figure
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure