On Constrained Minimum Pseudotriangulations

Abstract

In this paper, we show some properties of a pseudotriangle and present three combinatorial bounds: the ratio of the size of minimum pseudotriangulation of a point set S and the size of minimal pseudotriangulation contained in a triangulation T, the ratio of the size of the best minimal pseudotriangulation and the worst minimal pseudotriangulation both contained in a given triangulation T, and the maximum number of edges in any settings of S and T. We also present a linear-time algorithm for finding a minimal pseudotriangulation contained in a given triangulation. We finally study the minimum pseudotriangulation containing a given set of non-crossing line segments.

Research by Günter Rote was partly supported by the Deutsche Forschungsgemeinschaft (DFG) under grant RO 2338/2-1.

The work of Cao An Wang is supported by NSERC grant OPG0041629.

Lusheng Wang is fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 1087/00E].

The work of Yinfeng Xu is supported by NSFC(19731001) and NSFC(70121001) for excellent group project. The initial collaboration between Yinfeng Xu and Günter Rote on this paper was sponsored by the graduate program “Graduiertenkolleg Algorithmische Diskrete Mathematik” of the Deutsche Forschungsgemeinschaft (DFG), grant GRK219/3, when Yinfeng Xu visited Berlin.