The structural topology optimization approach can be used to generate compliant mechanisms for some desired input–output requirements. The success of the optimization depends on the structural geometry representation scheme used. In this paper, a novel representation scheme is proposed. The representation scheme is characterized by pairs of curves that are used to connect Input/Ouput (I/O) regions of the structure. Each pair of curves includes a normal curve and a fat curve. The areas bounded by the pair of curves define the material distribution between them. All I/O regions are connected to one another (either directly or indirectly) by pairs of curves in order to form one single connected load-bearing structure. A genetic algorithm for constrained and multiobjective optimization is then applied with the representation scheme of the structure in the form of a graph. Simulation results from a displacement inverter and a displacement redirector indicate that the presented representation scheme is appropriate.