Gabriel Graphs in Arbitrary Metric Space and their Cellular Automaton for Many Grids

Abstract

LUIDNEL MAIGNAN, INRIA Saclay FRÉDÉRIC GRUAU, INRIA Saclay, LRI, LIRMM Gabriel graphs are subgraphs of Delaunay graphs that are used in many domains such as sensor networks and computer graphics. Although very useful in their original form, their definition is bounded to applications involving Euclidean spaces only, but their principles seem to be applicable to a wider range of applications. In this article, we generalize this construct and define metric Gabriel graphs that transport the principles of Gabriel graphs on arbitrary metric space, allowing their use in domains like cellular automata and amorphous computing, or any other domains where a non-Euclidean metric is used. We study global/local properties of metric Gabriel graphs and use them to design a cellular automaton that draws the metric Gabriel graph of its input. This cellular automaton only uses seven states to achieve this goal and has been tested on hexagonal grids, 4-connected, and 8-connected square grids.

Hidden order in amorphous structures: Extraction of nearest neighbor networks of amorphous Nd–Fe alloys with Gabriel graph analyses

“…For many decades, the structural properties of amorphous systems have been widely studied [18][19][20][21][22] in terms of statistical approaches such as classical molecular dynamics (MD). Out of those studies, many characteristic functions and parameters (e. g. radial distribution functions, angular distribution functions and bond orientational order parameters) have been calculated as statistical values of amorphous systems to describe the structural properties of amorphous systems [23].…”