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One of the useful approaches to exploit redundancy in a sensor network is to actively keep only a small subset of sensors that are sufficient to cover the region required to be monitored. The set of active sensors should also form a connected communication graph, so that they can autonomously respond to application queries and/or tasks. Such a set of active sensors is known as a connected sensor cover, and the problem of selecting a minimum connected sensor cover has been well studied when the transmission radius and sensing radius of each sensor is fixed. In this article, we address the problem of selecting a minimum energy-cost connected sensor cover, when each sensor node can vary its sensing and transmission radius; larger sensing or transmission radius entails higher energy cost. For the above problem, we design various centralized and distributed algorithms, and compare their performance through extensive experiments. One of the designed centralized algorithms (called CGA) is shown to perform within an O(log n) factor of the optimal solution, where n is the size of the network. We have also designed a localized algorithm based on Voronoi diagrams which is empirically shown to perform very close to CGA, and due to its communication-efficiency, results in significantly prolonging the network lifetime.