On numbers of pseudo-triangulations

We study the maximum numbers of pseudo-triangulations and pointed pseudotriangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation. We derive the bounds O(5.45 ) and Ω(2.41 ) for the maximum number of pointed pseudo-triangulations that can be contained in a specific triangulation over a set of N points. For the number of all pseudo-triangulations contained in a triangulation we derive the bounds O∗(6.54N ) and Ω(3.30 ). We also prove that O∗(89.1N ) pointed pseudo-triangulations can be embedded over any specific set of N points in the plane, and at most 120 general pseudo-triangulations.