We study unit disk visibility graphs, where the relation between a pair of geometric entities is defined by not only obstacles, but also distance them. This particular graph class models real world scenarios more accurately compared to conventional graphs. first define and classify then show that 3-coloring problem NP-complete when model used for set line segments (which applies points) polygon...

Given a set D of unit disks in the Euclidean plane, we consider (i) the discrete unit disk cover (DUDC) problem and (ii) the rectangular region cover (RRC) problem. In the DUDC problem, for a given set P of points the objective is to select minimum cardinality subset D∗ ⊆ D such that each point in P is covered by at least one disk in D∗. On the other hand, in the RRC problem the objective is to...

Let $C_0(alpha)$ denote the class of concave univalent functions defined in the open unit disk $mathbb{D}$. Each function $f in C_{0}(alpha)$ maps the unit disk $mathbb{D}$ onto the complement of an unbounded convex set. In this paper, we study the mapping properties of this class under integral operators.

let $c_0(alpha)$ denote the class of concave univalent functions defined in the open unit disk $mathbb{d}$. each function $f in c_{0}(alpha)$ maps the unit disk $mathbb{d}$ onto the complement of an unbounded convex set. in this paper, we study the mapping properties of this class under integral operators.

A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of unit radius disks in the plane. We give upper and lower bounds on the number of labelled unit disk and disk graphs on n vertices. We show that the number of unit disk graphs on n vertices is n · α(n) and the number of disk graphs on n vertices is n · β(n), where α(n) and β(n) are Θ(1...

We characterize the complexity of a number of basic optimization problems for unit disk graphs speciied hierarchically as in BOW83, LW87a, Le88, LW92]. Both PSPACE-hardness results and polynomial time approximations are presented for most of the problems considered. These problems include minimum vertex coloring, maximum independent set, minimum clique cover, minimum dominating set and minimum ...

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 for and . the ...