In this paper we study a problem occuring in the context of geometric routing algorithms for mobile ad-hoc networks: Finding unit disk graph coordinates given a graph G = (V, E). Based on a proof that recognition of unit disk graphs is an NP-hard problem, we show that the problem of finding unit disk graph coordinates given a graph G = (V, E) is NPhard. Subsequently, we show that the proof does...

We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree db 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism i...

Let denote the unit circle in the complex plane. Given a function , one uses t usual (harmonic) Poisson kernel for the unit disk to define the Poisson integral of , namely . Here we consider the biharmonic Poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . We then consider the dilations ...