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We propose a novel localized topology-control algorithm for each wireless node to locally select communication neighbors and adjust its transmission power accordingly such that all nodes together self-form a topology that is energy efficient simultaneously for both unicast and broadcast communications. We theoretically prove that the proposed topology is planar, which meets the requirement of certain localized routing methods to guarantee packet delivery; it is power-efficient for unicast - the energy needed to connect any pair of nodes is within a small constant factor of the minimum; it is also asymptotically optimum for broadcast - the energy consumption for broadcasting data on top of it is asymptotically the best among all structures constructed using only local information; it has a constant bounded logical degree, which will potentially save the cost of updating routing tables if used. We further prove that the expected average physical degree of all nodes is a small constant. To the best of our knowledge, this is the first localized topology-control strategy for all nodes to maintain a structure with all these desirable properties. Previously, only a centralized algorithm was reported. Moreover, by assuming that the node ID and its position can be represented in O(log n) bits for a wireless network of n nodes, the total number of messages by our methods is in the range of theoretical results are corroborated in the simulations.