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 $$k\ge 0$$, does G have a strict-orthogonal drawing with at most k reflex angles per face? For $$k=0$$ the problem is equivalent to realizing each face as a rectangle. The problem can be reduced to a max-flow problem in some linear-size nonplanar network, but the best solutions require $$\varOmega n^{1.5} \log n\log k$$ time. We describe a graph matching approach that can decide strict-orthogonal drawability for arbitrary reflex complexity k in $$Onk^{1.5}$$ time, which is faster for constant values of k. 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.

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