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Improving the lifetime of sensor networks that are used to monitor a number of targets lying on a field has lately received considerable attention. The most common approach to tackle this problem divides the sensor nodes into cover sets, where the sensors in each set are capable of monitoring all targets in the field. By activating one cover set at a time, the sensor network lifetime can be extended. In this paper, we analyze the problem of connected partial target coverage where cover sets are allowed to monitor a subset of the targets at any point in time, while connectivity with the base station is retained. We propose an algorithm that computes the desired cover sets and we compare its performance, in terms of network lifetime, to that of a full coverage algorithm. Our solution takes into account several characteristics of the problem, such as the specific topology parameters, the coverage status of the sensors as well as their remaining battery life. Extensive experimental results are presented for different sensor and target placement topologies. Results show that monitoring 90% of the targets may yield twice the network lifetime provided by a full coverage approach.