ar X iv : m at h / 06 01 74 7 v 2 [ m at h . C O ] 1 4 Ju n 20 07 On the Number of Pseudo - Triangulations of Certain Point Sets ⋆

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12nnΘ(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, the Institut für Softwaretechnologie, Technische Universität Graz and the Departamento de Matemáticas, Universidad de Alcalá. Research of Oswin Aichholzer is partially supported by Acciones Integradas 20032004, Proj.Nr.1/2003 and by the FWF [Austrian Fonds zur Förderung der Wissenschaftlichen Forschung] under grant S09205-N12, FSP Industrial Geometry. Research of David Orden is partially supported by grants HU2002-0010, MTM200508618-C02-02 and S-0505/DPI/0235-02. Research of Francisco Santos is partially supported by Acción Integrada España-Austria HU2002-0010 and grant MTM200508618-C02-02 of Spanish Dirección General de Investigación Cient́ıfica. Email addresses: [email protected] (Oswin Aichholzer), [email protected] (David Orden), [email protected] (Francisco Santos), [email protected] (Bettina Speckmann). Preprint submitted to Elsevier 14 June 2007