2-level quasi-planarity or how caterpillars climb (SPQR-)trees

Given a bipartite graph G = (Vb, Vr, E), the 2-Level Quasi-Planarity problem asks for the existence of a drawing of G in the plane such that the vertices in Vb and in Vr lie along two parallel lines ℓb and ℓr, respectively, each edge in E is drawn in the unbounded strip of the plane delimited by ℓb and ℓr, and no three edges in E pairwise cross.