Local properties of geometric graphs

We introduce a new property for geometric graphs: the local diameter. This property is based on the use of the region counting distances introduced by Demaine, Iacono and Langerman as a generalization of the rank difference. We use it as a characterization of the local density of the vertices. We study the properties of those distances, and show that there is a strong relation between region counting distances and speed distances. We define the local diameter as the upper bound on the length of the shortest path between any pair of vertices, expressed as a function of the number of vertices in their neighborhood. We determine the local diameter of several well-studied graphs such as the Ordered Θ-graph and the Skip List Spanner. We also analyze the impact of the local diameter for different applications, such as path and point queries on geometric graphs.