On locally Delaunay geometric graphs

A geometric graph is a simple graph <i>G=(V,E)</i> with an embedding of the set <i>V</i> in the plane such that the points that represent <i>V</i> are in general position. A geometric graph is said to be <i>k-locally Delaunay</i> (or a <i>D</i>_k-graph) if for each edge <i>(u,v)</i> ∈ <i>E</i> there is a (Euclidean) disc <i>d</i> that contains <i>u</i> and <i>v</i> but no other vertex of <i>G</i> that is within <i>k</i> hops from <i>u</i> or <i>v</i>.The study of these graphs was recently motivated by topology control for wireless networks [6,7]. We obtain the following results: (i) We prove that if <i>G</i> is a <i>D</i>₁-graph on <i>n</i> vertices, then it has <i>O</i>(<i>n</i>^3/2) edges. (ii) We show that for any <i>n</i> there exist <i>D</i>₁-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 arrangement of $n$ unit circles in the plane is O(n^3/2).The first two results improve the best previously known upper and lower bounds of $O(n^ 5/3 )$ and $\Omega(n)$ respectively (see \cite KL03 ). 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(n^3/2 κ(n)) (see [1] ), where κ(n) = (log n ) ^O(α²(n)) and where α(n) is the extremely slowly growing inverse Ackermann's function.