Computational Geometry is a relatively young and very active field of research in the intersection of mathematics and theoretical computer science. Studying algorithms and data structures
have been main objectives of this growing discipline. Although
geometric graphs are structures defined by geometric properties, like
x- and y-coordinates, they have a highly discrete nature. Straight
lines spanned by a finite set of discrete points give rise to simple
and memory efficient data structures. While not loosing the geometric
information, geometric graphs additionally provide combinatorial
context (like neighborhood information) that is sufficient for many
applications and allows for very efficient and stable
algorithms. Moreover, for many problems the geometric information is
not needed for their solution. In these cases, point sets, geometric
in principle, can be stored and used in a purely combinatorial way. A
simple example is the construction of the convex hull of a point set,
which is an intrinsic task of countless algorithms. For this it is
sufficient to know for any triple a,b,c of points whether c is to the
left or to the right of the straight line spanned by a and b. A data
structure that stores this information is the so called order
type.
Not to be forced to rely on geometric information
has one major advantage: It enables for simple, exact, and robust
algorithms. For these reasons, Computational Geometry has become
highly interweaved with fields of Discrete Geometry like Combinatorial
Geometry. In the proposed project we want to explore a group of
interrelated questions that can be reduced to purely combinatorial
problems.
One exception from this group of purely
combinatorial problems is the question of blocking Delaunay
triangulations on bi-colored point sets. The order type does not
provide the Delaunay property for quadruples of points, an in-circle
property needed for Delaunay triangulation construction. An extended
order type, for instance, a “Delaunay order type” mapping
the Delaunay property to purely combinatorial data, could help solving
this and many other problems on Delaunay triangulations by answering
how many different Delaunay triangulation exist for a given
“classical” order type.
But even though not of pure combinatorial nature,
this subproblem is related to the other proposed problems. General
methods on bi-colored point sets can be applied to the problems on
compatible geometric graphs, isomorphic plane geometric graphs,
questions on k-convexity, and also, of course, the Erdős-Szekeres
type problems on bi-colored point sets. Further, new insights and
results on any of these problems will have implications on the whole
project and also to many other problems from Discrete & Computational
Geometry.
In the context of this project, examples for
interesting classes of geometric graphs are triangulations,
pseudo-triangulations, spanning trees, spanning circles, spanning
paths, and (perfect) matchings. As already mentioned, the proposed
problems are interrelated parts of one project. It is our strong
belief that attacking these problems in a combined attempt will have
synergetic effects to all parts. This will help to make considerable
progress on the presented questions, to gain additional insight into
their structure, and to finally answer at least some of them.