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Thermal management is an important challenge in modern electronics, avionics, automotive, and energy storage systems. While passive thermal solutions (like heat sinks or heat spreaders) are often used, actively modulating heat flow (e.g. via thermal switches or diodes) would offer additional degrees of control over the management of thermal transients and system reliability. Here we report the first thermal switch based on a flexible, collapsible graphene membrane, with low operating voltage, < 2 V. We also employ active-mode scanning thermal microscopy (SThM) to measure the device behavior and switching in real time. A compact analytical thermal model is developed for the general case of a thermal switch based on a double-clamped suspended membrane, highlighting the thermal and electrical design challenges. System-level modeling demonstrates the thermal trade-offs between modulating temperature swing and average temperature as a function of switching ratio. These graphene-based thermal switches present new opportunities for active control of fast (even nanosecond) thermal transients in densely integrated systems.