We consider the destructive effects of encounters between binaries and red giant stars in the Galactic Centre. Such encounters may explain the observed depletion of luminous red giants within the central 0.2pc of the galaxy. We consider encounters involving 2- and 8-Msolar red giants, and thus span the range of stellar masses contributing to the most luminous red giants observed in the Galactic Centre. To explore the phase space of encounters thoroughly, we simulate 18x10^3 encounters using a modified four-body code in which the red giant core and components of the binary are treated as point masses, and where the envelope configuration is assumed to remain static throughout the encounter. We then rerun a small number of encounters with a smoothed particle hydrodynamics (SPH) code to confirm the reliability of conclusions drawn from the four-body runs. We see two possible pathways to red giant destruction. A large fraction of encounters lead to the formation of common-envelope systems, where two compact objects (drawn from the red giant core and the components of the original binary) form a binary within a common gaseous envelope, whilst the third body is ejected. The destruction of the red giant will then follow when the envelope is ejected as the binary hardens. In a smaller number of encounters, the intruding binary passes through the star and ejects the red giant core from the envelope. The red giant envelope will then disperse on short time-scales. We compute the time-scales for both of these processes to occur in the Galactic Centre for a variety of binary populations.