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Optical saturable absorption through a monolayer graphene vertically inserted between fibers and overlaid on an ion-exchanged waveguide was measured for picosecond switching application. In our proposed switch, complete switching is performed by controlling the signal light amplitude with the configuration of two cascaded Mach-Zehnder interferometers. To control the signal light amplitude, we use saturable absorption in graphene. For switching, the modulation depth of the amplitude is required to be 2.414, which corresponds to 7.65 dB in intensity. The modulation depth of absorption larger than 10 dB was experimentally obtained in the graphene-loaded waveguide with length of 7 mm.