The maximal number of pairwise communicating stations under limitations of maximal and minimal communication distance

We consider the problem of determining the maximal number of stations that can maintain a total network of communication. We assume that there is a distance R which is that maximal distance that two stations can be separated and remain in contact. We also assume that there is a distance r which is them minimal separation that allows communication. This problem is intimately related to the problem of packing disks within a circle. The problem of finding the circle of smallest radius enclosing a finite set of points in the plane arises in a number of applications. Many numerical codes have been written for this problem. We provide a framework for investigating the geometric properties of this circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing a finite set of points P is the minimal circle if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We use this result to find a sharp estimate on the diameter of the minimal circle in terms of the diameter of P. We also show that the center of the minimal circle is contained in the convex hull of P.