On growing connected beta-skeletons

TL;DR

This paper introduces methods to grow connected beta-skeletons, a type of proximity graph, ensuring connectivity for any beta value and exploring their morphological transformations, with potential applications in biology and chemistry.

Contribution

The paper presents a novel approach to grow stable, connected beta-skeletons for any beta, addressing the typical disconnection issue at higher beta values.

Findings

Stable beta-skeletons can be grown for any beta value.

Morphological transformations depend on beta and approximation degree.

Potential applications in biological and chemical systems.

Abstract

A $\beta$-skeleton, $\beta \geq 1$, 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 $\beta$ determines the size and shape of the lune-based neighbourhood. A $\beta$-skeleton of a random planar set is usually a disconnected graph for $\beta>2$. With the increase of $\beta$, the number of edges in the $\beta$-skeleton of a random graph decreases. We show how to grow stable $\beta$-skeletons, which are connected for any given value of $\beta$ and characterise morphological transformations of the skeletons governed by $\beta$ and a degree of approximation. We speculate how the results obtained can be applied in biology and chemistry.