On locally Delaunay geometric graphs

A geometric graph is a simple graph G = (V, E) with an embedding of the set V in the plane such that the points that represent V are in general position. A geometric graph is said to be k-locally Delaunay (or a Dk-graph) if for each edge (u, v) ∈ E there is a (Euclidean) disc d that contains u and v but no other vertex of G that is within k hops from u or v. 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 G is a D1-graph on n vertices, then it has O(n3/2 ) edges. (ii) We show that for any n there exist D1-graphs with n vertices and Ω(n4/3 ) 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(n3/2 ). The first two results imp...

• Type: Conference
• Year: 2004
• Where: COMPGEOM
• Authors: Rom Pinchasi, Shakhar Smorodinsky
• Keywords: COMPGEOM 2004, Geometric Graph, Unit Circles, Upper Bound