Locally Delaunay Realizability of Regular Series-Parallel Graphs

This paper studies the combinatorial properties of geometric structures, known as k-locally Delaunay graphs, introduced in the application of wireless ad hoc networks. Given a network that models the connections among its sensors as a k-locally Delaunay graph, we investigate which topologies are allowed for such a network. We show that the family of regular series-parallel graphs, a subfamily of two-terminal series parallel graphs, can be realized as k-locally Delaunay graphs for any value of k. More precisely, for any regular series-parallel graph, we present a linear time algorithm for constructing the corresponding k-locally Delaunay graph for any value of k.