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This paper presents a class of routing protocols for vehicular ad hoc networks (VANETs) called the Intersection-based Geographical Routing Protocol (IGRP), which outperforms existing routing schemes in city environments. IGRP is based on an effective selection of road intersections through which a packet must pass to reach the gateway to the Internet. The selection is made in a way that guarantees, with high probability, network connectivity among the road intersections while satisfying quality-of-service (QoS) constraints on tolerable delay, bandwidth usage, and error rate. Geographical forwarding is used to transfer packets between any two intersections on the path, reducing the path's sensitivity to individual node movements. To achieve this, we mathematically formulate the QoS routing problem as a constrained optimization problem. Specifically, analytical expressions for the connectivity probability, end-to-end delay, hop count, and bit error rate (BER) of a route in a two-way road scenario are derived. Then, we propose a genetic algorithm to solve the optimization problem. Numerical and simulation results show that the proposed approach gives optimal or near-optimal solutions and significantly improves VANET performance when compared with several prominent routing protocols, such as greedy perimeter stateless routing (GPSR), greedy perimeter coordinator routing (GPCR), and optimized link-state routing (OLSR).