We consider a dense wireless sensor network where the radio transceivers of the sensor nodes are heavily duty-cycled in order to conserve energy. The chief purpose of the sensor network is surveillance and monitoring, where upon observation of certain event of interest, a sensor node generates a message and forwards it to a gateway located somewhere in or near the network. This forwarding relies on routes constructed using sensors whose radios are on/active. In order for such messages to reach the gateway with minimal delay, any sensor in the network should ideally have a route to the gateway consisting of active sensors at all times. Prior approaches to similar problems include clustering, virtual backbone, and connected dominating sets. Low energy consumption and good connectivity are potentially conflicting objectives. Our principal goal is to find an approach that results in the lowest possible duty cycle, and that provides better trade-off between the two objectives. In this paper we introduce the concept of partial clustering, which may be viewed as a generalized method of clustering. We compare the theoretical performance of different instances of partial clustering to that of standard clustering, and show that partial clustering can achieve lower duty cycle and provide greater flexibility in the trade-off between energy efficiency and connectivity. We then present a distributed algorithm based on the partial clustering method. Simulation results are provided to evaluate their effectiveness and energy efficiency.