Generalized $\beta$-skeletons

TL;DR

This paper introduces a generalized definition of $eta$-skeletons based solely on distance, enabling their application to weighted graphs and diverse objects, and presents new algorithms for constructing these graphs.

Contribution

It proposes a more general, distance-based definition of $eta$-skeletons, extending their applicability and establishing relations with other geometric graphs.

Findings

Unified framework for $eta$-skeletons and related graphs

New algorithms for constructing $eta$-skeletons

Applicability to weighted graphs and non-point objects

Abstract

$\beta$-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than the present one, definition of $\beta$-skeletons based only on the distance criterion. It allows us to consider them in many different cases, e.g. for weighted graphs or objects other than points. Two types of $\beta$-skeletons are especially well-known: the Gabriel Graph (for $\beta = 1$) and the Relative Neighborhood Graph (for $\beta = 2$). The new definition retains relations between those graphs and the other well-known ones (minimum spanning tree and Delaunay triangulation). We also show several new algorithms finding $\beta$-skeletons.