175 research outputs found
Point sets that minimize -edges, 3-decomposable drawings, and the rectilinear crossing number of
There are two properties shared by all known crossing-minimizing geometric
drawings of , for a multiple of 3. First, the underlying -point set
of these drawings has exactly -edges, for all . Second, all such drawings have the points divided into three
groups of equal size; this last property is captured under the concept of
3-decomposability. In this paper we show that these properties are tightly
related: every -point set with exactly -edges for
all , is 3-decomposable. As an application, we prove that the
rectilinear crossing number of is 9726.Comment: 14 page
Quasi-infra-red fixed points and renormalisation group invariant trajectories for non-holomorphic soft supersymmetry breaking
In the MSSM the quasi-infra-red fixed point for the top-quark Yukawa coupling
gives rise to specific predictions for the soft-breaking parameters. We discuss
the extent to which these predictions are modified by the introduction of
additional ``non-holomorphic'' soft-breaking terms. We also show that in a
specific class of theories there exists an RG-invariant trajectory for the
``non-holomorphic'' terms, which can be understood using a holomorphic spurion
term.Comment: 24 pages, TeX, two figures. Uses Harvmac (big) and epsf. Minor errors
corrected, and the RG trajectory explained in terms of a holomorphic spurion
ter
Balanced Islands in Two Colored Point Sets in the Plane
Let be a set of points in general position in the plane, of which
are red and of which are blue. In this paper we prove that there exist: for
every , a convex set containing
exactly red points and exactly
blue points of ; a convex set containing exactly red points and exactly blue points of . Furthermore, we present
polynomial time algorithms to find these convex sets. In the first case we
provide an time algorithm and an time algorithm in the
second case. Finally, if is
small, that is, not much larger than , we improve the running
time to
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
Embedding Four-directional Paths on Convex Point Sets
A directed path whose edges are assigned labels "up", "down", "right", or
"left" is called \emph{four-directional}, and \emph{three-directional} if at
most three out of the four labels are used. A \emph{direction-consistent
embedding} of an \mbox{-vertex} four-directional path on a set of
points in the plane is a straight-line drawing of where each vertex of
is mapped to a distinct point of and every edge points to the direction
specified by its label. We study planar direction-consistent embeddings of
three- and four-directional paths and provide a complete picture of the problem
for convex point sets.Comment: 11 pages, full conference version including all proof
Flip Graphs of Degree-Bounded (Pseudo-)Triangulations
We study flip graphs of triangulations whose maximum vertex degree is bounded
by a constant . In particular, we consider triangulations of sets of
points in convex position in the plane and prove that their flip graph is
connected if and only if ; the diameter of the flip graph is .
We also show that, for general point sets, flip graphs of pointed
pseudo-triangulations can be disconnected for , and flip graphs of
triangulations can be disconnected for any . Additionally, we consider a
relaxed version of the original problem. We allow the violation of the degree
bound by a small constant. Any two triangulations with maximum degree at
most of a convex point set are connected in the flip graph by a path of
length , where every intermediate triangulation has maximum degree
at most .Comment: 13 pages, 12 figures, acknowledgments update
Quadratic-time, linear-space algorithms for generating orthogonal polygons with a given number of vertices
Programa de Financiamento Plurianual, Fundação para a Ciéncia e TecnologiaPrograma POSIPrograma POCTI, FCTFondo Europeo de Desarrollo Regiona
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