 
Summary: On Minimum Weight PseudoTriangulations
Oswin Aichholzer Franz Aurenhammer Thomas Hackl Bettina Speckmann§
Abstract
In this note we discuss some structural properties of minimum weight pseudotriangulations.
1 Introduction
Optimal triangulations for a set of points in the plane have been, and still are, extensively studied
within Computational Geometry. There are many possible optimality criteria, often based on
edge weights or angles. One of the most prominent criteria is the weight of a triangulation, that
is, the total Euclidean edge length. Computing a minimum weight triangulation (MWT) for a
point set has been a challenging open problem for many years [4] and various approximation
algorithms were proposed over time; see e.g. [3] for a short survey. Mulzer and Rote [9] showed
only very recently that the MWT problem is NPhard.
Pseudotriangulations are related to triangulations and use pseudotriangles in addition to
triangles. A pseudotriangle is a simple polygon with exactly three interior angles smaller
than . Also for pseudotriangulations several optimality criteria have been studied, for example,
concerning the maximum face or vertex degree [5]. Optimal pseudotriangulations can also be
found via certain polytope representations [10] or via a realization as locally convex surfaces in
threespace [1]. Not all of these optimality criteria have natural counterparts for triangulations.
Here we consider the classic minimum weight criterion for pseudotriangulations.
Rote et al. [11] were the first to ask for an algorithm to compute a minimum weight pseudo
