Hannes Krasser

On the Crossing Number of Complete Graphs

Let (G) denote the rectilinear crossing number of a graph G. We determine (K11)=102 and (K12)=153. Despite the remarkable hunt for crossing numbers of the complete graph Kn – initiated by R. Guy in the 1960s – these quantities have been unknown forn>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11.Based on these findings, we establish a new upper bound on (Kn) for general n. The bound stems from a novel construction of drawings of Kn with few crossings.

Farthest line segment Voronoi diagrams

The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(n log n) time construction algorithm that is easy to implement. No restrictions are placed upon the n input line segments; they are allowed to touch or cross.

Pseudotriangulations from Surfaces and a Novel Type of Edge Flip

We prove that planar pseudotriangulations have realizations as polyhedral surfaces in three-space. Two main implications are presented. The spatial embedding leads to a novel flip operation that allows for a drastic reduction of flip distances, especially between (full) triangulations. Moreover, several key results for triangulations, like flipping to optimality, (constrained) Delaunayhood, and a convex polytope representation, are extended to pseudotriangulations in a natural way.

On the number of plane graphs

We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide simple proofs that the number of spanning trees, cycle-free graphs (forests), perfect matchings, and spanning paths is also minimized for point sets in convex position.In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*(√72^<i>n</i>) = Θ*(8.4853^<i>n</i>) triangulations and Θ*(41.1889^<i>n</i>) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.

Enumerating order types for small sets with applications

Order types are a means to characterize the combinatorial properties of a finite point configuration. In particular, the crossing properties of all straight-line segments spanned by an planar

Convexity minimizes pseudo-triangulations

n

On the crossing number of complete graphs

-point set are reflected by its order type. We establish a complete and reliable data base for all possible order types of size

Evolution strategy and hierarchical clustering

n=10

Towards compatible triangulations

or less. The data base includes a realizing point set for each order type in small integer grid representation. To our knowledge, no such project has been carried out before. We substantiate the usefulness of our data base by applying it to several problems in computational and combinatorial geometry. Problems concerning triangulations, simple polygonalizations, complete geometric graphs, and

Adapting (Pseudo)-Triangulations with a Near-Linear Number of Edge Flips

k