FWF - OTECG - Order Types and Extremal Combinatorial Geometry

We propose a research project in the field of computational geometry, a sub-field of mathematics and theoretical computer science that focuses on the theoretical backgrounds of how computer programs can process geometric entities. Our research has applications in extremal combinatorial geometry, where we are interested in estimating the numbers of different “networks” defined by straight-line connections drawn between sites (represented by points) on a flat surface, similar to straight routes connecting cities on a map. Such drawings are called geometric graphs. Geometric graphs can be classified in various ways, for example whether or not they contain crossings or cycles. For many classes without crossings, it is known that the number of different geometric graphs (obtained by connecting given points) is minimized when the points are placed on a circle. Analogously, arranging the points in that way maximizes the number of such graphs for many geometric graph classes with crossings. One objective of our work is to characterize point sets that maximize or minimize the number of geometric graphs for certain classes. For such problems, we are usually not interested in the exact position of the points, but how they are placed relative to each other, since, in general, our problem does not change if we slightly move a few points. The “order type” of a point set tells us in which order we encounter three points of the set when walking along the boundary of the triangle defined by these points in counterclockwise direction; that is, we get the orientation of each triple of points. In previous work, we defined a relation on order types by comparing the set of crossing-free geometric graphs each order type admits. We characterized order types among which there are point sets that have the most or the least number of certain classes of geometric graphs. However, this characterization is rather general. We plan to obtain relations between order types that provide more insight into the structure of maximizing and minimizing point sets. In particular, we are interested in bounding the number of triangulations (geometric graphs in which each bounded area is a triangle). We will attempt to prove a longstanding conjecture stating that the number of triangulations is minimized by a certain order type, using the insights obtained from different relations and local transformations (for example changing the orientation of a single point triple). Apart from bounding the number of geometric graphs, we are interested in relations between different order types in its own right. We will analyze different types of local operations that allow transforming one order type into another.