Sensor networks and covering of plane by discs

The problem of power-efficient regular covering of a plane region by sensors with two adjustable sensing ranges is considered. We improve and strengthen the previously known results on covers' properties as well as optimize the total energy consumption for sensing and data transmission between the elements of the sensor network. From the mathematical point of view, we formulate and solve several problems of covering the plane by discs as in the classical papers of L. F. Toth about arrangement on the plane. The considered problems relate to special types of covers but give some presentation of the general case. It is important that adjustability of the radii of discs turns out actual and prospective that the applied view point. Il. 8, bibl. 12.