Abstract Let G be a bipartite graph, and let λ_e,λ_i be two parallel convex curves; we study the question about whether G admits a planar straight-line drawing such that the vertices of one partite set of G lie on λ_e and the vertices of the other partite set lie on λ_i. A characterization is presented that gives rise to linear time testing algorithm. We also describe a drawing algorithm that runs in linear time if the curves are two concentric circles and the real RAM model of computation is adopted.