On the Complexity of Realizing Facial Cycles

We study the following combinatorial problem. Given a planar graph G = (V,E) and a set of simple cycles C in G, find a planar embedding E of G such that the number of cycles in C that bound a face in E is maximized. We establish a tight border of tractability for this problem in biconnected planar graphs by giving conditions under which the problem is NP-hard and showing that relaxing any of these conditions makes the problem polynomial-time solvable. Moreover, we give a 2-approximation algorithm for series-parallel graphs and a (4+ε)approximation for biconnected planar graphs.