Let m be a random tessellation in R, d ≥ 1, observed in the window Wρ = ρ1/d[0, 1], ρ > 0, and let f be a geometrical characteristic. We investigate the asymptotic behaviour of the maximum of f(C) over all cells C ∈ m with nucleus in Wρ as ρ goes to infinity. When the normalized maximum converges, we show that its asymptotic distribution depends on the so-called extremal index. Two examples of ...

Given an arbitrary convex shape C, a set P of points in the plane and a set S of line segments whose endpoints are in P , a constrained generalized Delaunay graph of P with respect to C denoted CDGC(P ) is constructed by adding an edge between two points p and q if and only if there exists a homothet of C with p and q on its boundary and no point of P in the interior visible to both p and q. We...

Graph theoretic properties of proteins can be used to perceive the differences between correctly folded proteins and well designed decoy sets. 3D protein structures of proteins are represented with graphs. We used two different graph representations: Delaunay tessellations of proteins and contact map graphs. Graph theoretic properties for both graph types showed high classification accuracy for...

A localized Delaunay triangulation owns the following interesting properties in a wireless ad hoc setting: it can be built with localized information, the communication cost imposed by control information is limited and it supports geographical routing algorithms that offer guaranteed convergence. This paper presents a localized algorithm that builds a graph called planar localized Delaunay tri...

We describe an algorithm that takes as input n points in the plane and a parameter , and produces as output an embedded planar graph having the given points as a subset of its vertices in which the graph distances are a (1 + )-approximation to the geometric distances between the given points. For point sets in which the Delaunay triangulation has sharpest angle α, our algorithm’s output has O( ...

In computer vision, we have the problem of creating graphs out of unstructured point-sets, i.e. the data graph. A common approach for this problem consists of building a triangulation which might not always lead to the best solution. Small changes in the location of the points might generate graphs with unstable configurations and the topology of the graph could change significantly. After buil...

Oblivious online routing (OOR) algorithms are suitable when applications only have local information available to make routing decisions. This paper presents two new OOR algorithms for Delaunay triangulations: the Ellipse Routing algorithm and the Arc Routing algorithm. Both of their names come from the shapes of their searching areas for the next-hop neighbor. This paper also evaluates and com...

This paper studies the construction of self-stabilizing topologies for distributed systems. While recent research has focused on chain topologies where nodes need to be linearized with respect to their identifiers, we go a step further and explore a natural 2-dimensional generalization. In particular, we present a local self-stabilizing algorithm that constructs a Delaunay graph from any initia...

Graph-theoretic properties of certain proximity graphs defined on planar point sets are investigated. We first consider some of the most common proximity graphs of the family of the Delaunay graph, and study their number of edges, minimum and maximum degree, clique number, and chromatic number. In the second part of the paper we focus on the higher order versions of some of these graphs and giv...

We consider two classes of higher order proximity graphs defined on a set of points in the plane, namely, the k-Delaunay graph and the k-Gabriel graph. We give bounds on the following combinatorial and geometric properties of these graphs: spanning ratio, diameter, chromatic number, and minimum number of layers necessary to partition the edges of the graphs so that no two edges of the same laye...