Before revision:  We map the problem of testing  the existence of such representations to the one of testing  the existence of suitably constrained 1-page book-embeddings and propose several polynomial-time or pseudo-polynomial-time algorithms.
After revision:  We map the problem of testing the existence of such representations to the one of testing the existence of suitably constrained 1-page book-embeddings and propose several polynomial-time and pseudo-polynomial-time algorithms.