Our goal is to save energy in wireless sensor networks (WSNs) by periodic duty-cycling of sensor nodes. We schedule sensor nodes between active (transmit or receive) and sleep modes while bounding packet latency in the presence of collisions. In order to support a dynamic WSN topology, we focus on topology-transparent approaches to scheduling; these are independent of detailed topology information. Much work has been done on topology-transparent scheduling in which all nodes are active. In this work, we examine the connection between topology-transparent duty-cycling and such non-sleeping schedules. This suggests a way to construct topology-transparent duty-cycling schedules. We analyse the performance of topology-transparent schedules with a focus on throughput in the worst case. A construction of topology-transparent duty-cycling schedules based on a topology-transparent non-sleeping schedule is proposed. The constructed schedule achieves the maximum average throughput in the worst case if the given non-sleeping schedule satisfies certain properties.