Topology control algorithms have been proposed to maintain network connectivity while improving energy efficiency and increasing network capacity. However, by reducing the number of links in the network, topology control algorithms actually decrease the degree of routing redundancy. As a result, the derived topology is more susceptible to node failures or departures. In this paper, we resolve this problem by enforcing k-vertex connectivity in the topology construction process. We propose a fully localized algorithm, fault-tolerant local spanning subgraph (FLSS), that can preserve k-vertex connectivity and is min-max optimal among all strictly localized algorithms (i.e., FLSS minimizes the maximum transmission power used in the network, among all strictly localized algorithms that preserve k-vertex connectivity). It can also be proved that FLSS outperforms two other existing localized algorithms in terms of reducing the transmission power. We also discuss how to relax several widely used assumptions in topology control to increase the practical utility of FLSS. Simulation results indicate that, compared with existing distributed/localized fault-tolerant topology control algorithms, FLSS not only has better power-efficiency, but also leads to higher network capacity. Moreover, FLSS is robust with respect to position estimation errors.