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Cooperative communication with single relay selection is a simple but effective communication scheme for energy-constrained networks. In this paper, we propose a novel selective single-relay cooperative scheme, combining selective-relay cooperative communication with physical-layer power control. Based on the MAC-layer RTS-CTS signaling, a set of potential relays compute individually the required transmission power to participate in the cooperative communication, and compete within a window of fixed length. The "best" relay is selected in a distributed fashion with minimum signaling overhead. We derive power-control solutions corresponding to two policies on relay selection: one is to minimize the energy consumption per data packet, and the other is to maximize the network lifetime. Our numerical and simulation results confirm that the proposed scheme achieves significant energy savings and prolongs the network lifetime considerably.