On the maximal number of edges of many faces in an arrangement
Journal of Combinatorial Theory Series A
Handbook of discrete and computational geometry
Geometric Graphs with No Self-intersecting Path of Length Three
GD '02 Revised Papers from the 10th International Symposium on Graph Drawing
Topological graphs with no self-intersecting cycle of length 4
Proceedings of the nineteenth annual symposium on Computational geometry
On levels in arrangements of curves
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Extremal Graph Theory
Policy controlled self-configuration in unattended wireless sensor networks
Journal of Network and Computer Applications
On the topologies of local minimum spanning trees
CAAN'06 Proceedings of the Third international conference on Combinatorial and Algorithmic Aspects of Networking
ICCSA'06 Proceedings of the 6th international conference on Computational Science and Its Applications - Volume Part I
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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 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(n3/2 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.