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Let T be a triangulation of a simple polygon. A flip in T is the operation of removing one diagonal of T and adding a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex… (More)

- Oswin Aichholzer, Thomas Hackl, +5 authors Emo Welzl
- CCCG
- 2014

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph GK n on any set S of n points in general position in the plane?

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe… (More)

- José Miguel Díaz-Báñez, Matias Korman, Pablo Pérez-Lantero, Alexander Pilz, Carlos Seara, Rodrigo I. Silveira
- Comput. Geom.
- 2013

We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that… (More)

- Alexander Pilz
- Comput. Geom.
- 2014

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally… (More)

Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erd˝ os-type question on the least number h k (n) of convex k-holes in S, and give improved lower bounds on

- Oswin Aichholzer, Ruy Fabila Monroy, +4 authors Birgit Vogtenhuber
- CCCG
- 2010

Given a set B of n blue points in general position, we say that a set of red points R blocks B if in the Delaunay triangulation of B ∪ R there is no edge connecting two blue points. We give the following bounds for the size of the smallest set R blocking B: (i) 3n/2 red points are always sufficient to block a set of n blue points, (ii) if B is in convex… (More)

- Oswin Aichholzer, Thomas Hackl, +4 authors Birgit Vogtenhuber
- Graphs and Combinatorics
- 2009

Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p ∈ S, if the parity of the degree of p in G matches its label. In this paper we study how well various classes of planar… (More)

- Oswin Aichholzer, Tillmann Miltzow, Alexander Pilz
- Comput. Geom.
- 2013

- Oswin Aichholzer, Matias Korman, Alexander Pilz, Birgit Vogtenhuber
- Algorithmica
- 2012

The geodesic between two points a and b in the interior of a simple polygon P is the shortest polygonal path inside P that connects a to b. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points… (More)