Proximity Structures for Geometric Graphs

In this paper we study proximity structures like Delauney triangulations based on geometric graphs, i.e. graphs which are subgraphs of the complete geometric graph. Given an arbitrary geometric graph G, we define several restricted Voronoi diagrams, restricted Delaunay triangulations, relative neighborhood graphs, Gabriel graphs and then study their complexities when G is a general geometric graph or G is some special graph derived from the application area of wireless networks. Besides being of fundamental interest these structures have applications in topology control for wireless networks.