How β-skeletons lose their edges

A β-skeleton is a proximity graphs with node neighbourhood defined by continuous-valued parameter β. Two nodes in a β-skeleton are connected by an edge if their lune-based neighbourhood contains no other nodes. With increase of β some edges a skeleton are disappear. We study how a number of edges in β-skeleton depends on β. We speculate how this dependence can be used to discriminate between random and non-random planar sets. We also analyse stability of β-skeletons and their sensitivity to perturbations.