On β-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 β > 1/sin κ, the β-skeleton of S is a subgraph of a minimum weight triangulation of S, where κ = tan^-1 (3/√2√3) ≈ π/3.1. There exists a four-point example such that the β-skeleton for β < 1/sin(π/3) is not a subgraph of the minimum weight triangulation.