On the number of pseudo-triangulations of certain point sets

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Abstract

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 12n nΘ(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.

Keywords

Pseudo-triangulations
Triangulations
Double-circle
Double-chain
Counting

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Parts of this work were done while the authors visited the Departamento de Matemática 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 España–Austria and by the FWF [Austrian Fonds zur Förderung der Wissenschaftlichen Forschung] under grant S09205-N12, FSP Industrial Geometry. Research of David Orden and Francisco Santos is partially supported by Acciones Integradas España–Austria and by grant MTM2005-08618-C02-02 of Spanish Dirección General de Investigacin Científica. David Orden is additionally supported by S0505/DPI/0235-02.