Orthogonal Graph Drawing with Flexibility Constraints

In this work we consider the following problem. Given a planar graph G with maximum degree 4 and a function flex E->N_o: that gives each edge a flexibility. Does G admit a planar embedding on the grid such that each edge e has at most flex(e) bends? Note that in our setting the combinatorial embedding of G is not fixed. We give a polynomial-time algorithm for this problem when the flexibility of each edge is positive. This includes as a special case the problem of deciding whether G admits a drawing with at most one bend per edge