### Description

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.

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.

Status | Finished |
---|---|

Effective start/end date | 1/09/11 → 31/12/15 |