While having high bandwidth-efficiency, the ad-hoc on-demand distance vector (AODV) routing protocol suffers from high signaling overhead due to route request (RREQ) messages flooding, especially when the node density and the number of connections are increased. In order to resolve this broadcast storm problem of the AODV in a high node density mobile ad-hoc network, we propose a geographical on-demand route discovery scheme. Assuming a known location of the destination, the RREQ of the proposed routing protocol is propagated in a unicast manner by employing a novel parsing mechanism for possible duplicate RREQs. The routing overhead of the proposed routing protocol is greatly robust to the node density change. We derive the node density required for the proposed routing protocol to keep the same connectivity as the AODV under the circumstance where the nodes are uniformly distributed. In addition, we present an imaginary destination consideration method to incorporate the uncertainty of the destination's location due to mobility. Computer simulations show that the proposed scheme enables the RREQ propagation to cover 95% of the one-hop communication area centered at the originally known location of the destination without sacrificing the unicast feature.