A fault-tolerant 1-spanner is used to preserve all the minimum energy paths after node failures to cope with fault-tolerant topology control problems in wireless ad hoc networks. A fault-tolerant 1-spanner is a graph such that the remaining graph after node failures will not only remain connected, but also have a stretch factor of one. The fault-tolerant 1-spanner is used in a localized and distributed topology control algorithm, named the k-Fault-Tolerant 1-Spanner (k-FT1S), where each node constructs a minimum energy path tree for every local failed node set. This paper proves that the topology constructed by k-FT1S is a k-fault-tolerant 1-spanner that can tolerate up to k node failures, such that the remaining network after node failures preserves all the minimum energy paths of the remaining network gained from the initial network by removing the same failed nodes. Simulations show that the remaining network after removal of any k nodes still has the optimal energy efficiency and is competitive in terms of average logical degree, average physical degree, and average transmission radius.
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