We study strategyproof (SP) mechanisms for the location of a facility on a discrete graph. We give a full characterization of SP mechanisms on lines and on sufficiently large cycles. Interestingly, the characterization deviates from the one given by Schummer and Vohra (2004) for the continuous case. In particular, it is shown that an SP mechanism on a cycle is close to dictatorial, but all agents can affect the outcome, in contrast to the continuous case. Our characterization is also used to derive a lower bound on the approximation ratio with respect to the social cost that can be achieved by an SP mechanism on certain graphs. Finally, we show how the representation of such graphs as subsets of the binary cube reveals common properties of SP mechanisms and enables one to extend the lower bound to related domains.