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In this paper, we present a localized algorithm for constructing power efficient topology for wireless ad hoc networks. Each mobile node determines its own transmission power based only on local information. The proposed algorithm first constructs the constrained Gabriel graph from the given unit disk graph and then reduces the total transmission power by allowing each node individually excises some replaceable links. The constructed topology is sparse, has a constant bounded power stretch factor, and the total transmission power is lower than those obtained from other proposed algorithms. In addition, compared with others, our algorithm requires lower time complexity to generate a solution, and can thus further save the energy for each mobile node. We demonstrate the performance improvements of our algorithm through simulations.