On growing connected beta-skeletons

A $β$-skeleton, $β\geq 1$, is a planar proximity undirected graph of an 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 $β>2$. 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.

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