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The topology of a wireless multi-hop network can be controlled by varying the transmission power at each node. In this paper, we give a detailed analysis of a cone-based distributed topology-control (CBTC) algorithm. This algorithm does not assume that nodes have GPS information available; rather it depends only on directional information. Roughly speaking, the basic idea of the algorithm is that a node u transmits with the minimum power pu,α required to ensure that in every cone of degree α around u, there is some node that u can reach with power pu,α. We show that taking α=5π/6 is a necessary and sufficient condition to guarantee that network connectivity is preserved. More precisely, if there is a path from s to t when every node communicates at maximum power then, if α≤5π/6, there is still a path in the smallest symmetric graph Gα containing all edges (u,v) such that u can communicate with v using power pu,α. On the other hand, if α>5π/6, connectivity is not necessarily preserved. We also propose a set of optimizations that further reduce power consumption and prove that they retain network connectivity. Dynamic reconfiguration in the presence of failures and mobility is also discussed. Simulation results are presented to demonstrate the effectiveness of the algorithm and the optimizations.