On the delaunay graph of a geometric graph

In this paper we study proximity structures for geometric graphs. The study of these structures was recently motivated by topology control for wireless networks [6, 7]. We obtain the following results: (i) We prove that if G is a D1-graph on n vertices, then it has O(n 3/2) edges. (ii) We show that for any n there exist D1-graphs with n vertices and Ω(n 4/3) edges. (iii) We prove that if G is a D2-graph on n vertices, then it has O(n) edges. This bound is worst-case asymptotically tight. As an application of the first result, we show that: (iv) The maximum size of a family of pairwise non-overlapping lenses in an arrange-ment of n unit circles in the plane is O(n3/2). The first two results improve the best previously known upper and lower bounds of O(n5/3) and Ω(n) respectively (see [6]). The third result improves the best previously known upper bound of O(n log n) ([6]). Finally, our last result improves the best previously known upper bound (for the more general case of not necessarily unit circles) of O(n3/2κ(n)) (see [1]), where κ(n) = (log n)O(α 2(n)) and where α(n) is the extremely slowly growing inverse Ackermann’s function.