ARTICLE IN PRESS COMGEO:969 Please cite this article in press as: O. Aichholzer et al., On minimum weight pseudo-triangulations, Computational Geometry (2008), doi:10.1016/j.comgeo.2008.10.002 JID:COMGEO AID:969 /FLA [m3G; v 1.14; Prn:6/11/2008; 10:24] P.1 (1-5) Computational Geometry ••• (••••) •••–••• Contents lists available at ScienceDirect Computational Geometry: Theory and Applications www.elsevier.com/locate/comgeo On minimum weight pseudo-triangulations Oswin Aichholzer a,1 , Franz Aurenhammer b,1 , Thomas Hackl a,1 , Bettina Speckmann c,∗ a Institute for Software Technology, Graz University of Technology, Graz, Austria b Institute for Theoretical Computer Science, Graz University of Technology, Graz, Austria c Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands article info abstract Article history: Received 19 February 2008 Received in revised form 3 October 2008 Accepted 17 October 2008 Communicated by F. Hurtado Keywords: Pseudo-triangulations Minimum weight In this note we discuss some structural properties of minimum weight pseudo-triangu- lations of point sets.  2008 Elsevier B.V. All rights reserved. 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 NP-hard. Pseudo-triangulations are related to triangulations and use pseudo-triangles in addition to triangles. A pseudo-triangle is a simple polygon with exactly three interior angles smaller than π . Also for pseudo-triangulations several optimality criteria have been studied, for example, concerning the maximum face or vertex degree [5]. Optimal pseudo-triangulations can also be found via certain polytope representations [10] or via a realization as locally convex surfaces in three-space [1]. Not all of these optimality criteria have natural counterparts for triangulations. Here we consider the classic minimum weight criterion for pseudo-trian gulations. Rote et al. [11] were the first to ask for an algorithm to compute a minimum weight pseudo-triangulation (MWPT). The complexity of the MWPT problem is unknown, but Levcopoulos and Gudmundsson [7] show that a 12-approximation of an MWPT can be computed in O(n 3 ) time. Moreover, they give an O(log n · w(MST)) approximation of an MWPT, in O(n log n) time. Here w(MST) is the weight of the minimum Euclidean spanning tree, which is a subset of the obtained structure. A pseudo-triangulation is called pointed (or minimum) if every vertex p has one incident region (either a pseudo-triangle or the exterior face) whose angle at p is greater than π . A pointed pseudo-triangulation minimizes the number of edges among all pseudo-triangulations of a given point set. Since a spanning tree is not necessarily pointed (see [2]) the pseudo- triangulation constructed by the approximation algorithm of [7] is also not necessarily pointed. It is logical to conjecture * Corresponding author. E-mail addresses: oaich@ist.tugraz.at (O. Aichholzer), auren@igi.tugraz.at (F. Aurenhammer), thackl@ist.tugraz.at (T. Hackl), speckman@win.tue.nl (B. Speckmann). 1 Supported by the Austrian FWF Joint Research Project ‘Industrial Geometry’ S9205-N12. 0925-7721/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.comgeo.2008.10.002