Orthogonal layout with optimal face complexity

We study a problem motivated by rectilinear schematization of geographic maps. Given a biconnected plane graph G and an integer k≥0, does G have a strict-orthogonal drawing (i.e., an orthogonal drawing without edge bends) with at most k reflex angles per face? For k=0, the problem is equivalent to realizing each face as a rectangle. We prove that the strict-orthogonal drawability problem for arbitrary reflex complexity k can be reduced to a graph matching or a network flow problem. Consequently, we obtain an O˜(n10/7k1/7)-time algorithm to decide strict-orthogonal drawability, where O˜(r) denotes O(rlogc⁡r), for some constant c. In contrast, if the embedding is not fixed, we prove that it is NP-complete to decide whether a planar graph admits a strict-orthogonal drawing with reflex face complexity 4.