On beta-skeleton as a subgraph of the minimum weight triangulation

Given a set S of n points in the plane, a triangulation is a maximal set of non-intersecting edges connecting the points in S. The weight of the triangulation is the sum of the lengths of the edges. In this paper, we show that for ¿ 1=sin , the -skeleton of S is a subgraph of a minimum weight triangulation of S, where = tan−1(3= √ 2 √ 3) ≈ =3:1. There exists a four-point example such that the -skeleton for ¡ 1=sin( =3) is not a subgraph of the minimum weight triangulation. c © 2001 Elsevier Science B.V. All rights reserved.