On growing connected β-skeletons

A β  -skeleton, β⩾1β⩾1, is a planar proximity undirected graph of a Euclidean points set, where nodes are connected by an edge if their lune-based neighbourhood contains no other points of the given set. Parameter β determines the size and shape of the lune-based neighbourhood. A β  -skeleton of a random planar set is usually a disconnected graph for β>2β>2. With the increase of β, the number of edges in the β-skeleton of a random graph decreases. We show how to grow stable β-skeletons, which are connected for any given value of β and characterise morphological transformations of the skeletons governed by β and a degree of approximation. We speculate how the results obtained can be applied in biology and chemistry.