Local properties of geometric graphs

Abstract

We propose a definition of locality for properties of geometric graphs. We measure the local density of graphs using region-counting distances between pairs of vertices, and we use this density to define local properties of classes of graphs. We illustrate locality by introducing the local diameter of geometric graphs: we define it as the upper bound on the size of the shortest path between any pair of vertices, expressed as a function of the density of the graph around these vertices. We determine the local diameter of several well-studied graphs such as Θ-graphs, Ordered Θ-graphs and Skip List Spanners. We also show that various operations, such as path and point queries using geometric graphs as data structures, have complexities which can be expressed as local properties.