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Connected-k-coverage has been recognized as an effective notion for prolonging the sensor network lifetime. It requires the monitored region to be k-covered by a connected component of active sensors, which is less demanding than requiring k-coverage and connectivity among all active sensors simultaneously. While critical conditions for k-coverage and connectivity have been studied separately before, there is a lack of fundamental understanding about connected-k-coverage. In this paper, as one of the first efforts in understanding the fundamental basics about connected-k-coverage, we analyze the critical conditions for connected-k-coverage using the percolation theorem and demonstrate their effectiveness using simulation results.