We study the number of crossings among edges of some higher order proximity graphs of the family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we give lower and upper bounds on the minimum and the maximum number of crossings that these geometric graphs defined on P have.

Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R) in a way that guarantees message delivery with greedy routing. It is well known that Delaunay triangulations are such embeddings. We present the algorithm FindAng...

Over 20 years ago, Dillencourt [1] showed that all outerplanar graphs can be realized as Delaunay triangulations of points in convex position. His proof is elementary, constructive and leads to a simple algorithm for obtaining a concrete Delaunay realization. In this note, we provide two new, alternate, also quite elementary proofs.

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable ...

We propose a new geometric spanner for wireless ad hoc networks, which can be constructed efficiently in a distributed manner. It integrates the connected dominating set and the local Delaunay graph to form a backbone of the wireless network. Priori arts showed that both structures can be constructed locally with bounded communication costs. This new spanner has these following attractive prope...

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs (p, r) where p is a point in the plane and r is a real number. The distance between two points (pi, ri) and (pj , rj) is defined as |pipj | − ri − rj . We show that in the case where all ri are positive numbers and |pipj | ≥ ri + rj for all i, j (in which case the po...

Circles are frequently used for modelling the growth of particle aggregates through the Voronoi diagram of circles, that is a special instance of the Johnson-Mehl tessellation. The Voronoi diagram of a set of sites is a decomposition of space into proximal regions. The proximal region of a site is the locus of points closer to that site than to any other one. Voronoi diagrams allow one to answe...

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) composites with ideally conducting spherical inclusions is established. The location optimal-design problem yields inclusions. geometrical-packing physical-conductivity problems are stated a periodic toroidal d-dimensional space an arbitrarily fixed number n nonoverlapping per periodicity...