Orthogonal Layout with Optimal Face Complexity: NP-hardness and Polynomial-time Algorithms

Given a biconnected plane graph G and a nonnegative integer k, we examine the problem of deciding whether G admits a strict-orthogonal drawing (i.e., an orthogonal drawing without bends) such that the reflex face complexity (the maximum number of reflex angles in any face) is at most k. We introduce a new technique to solve the problem in O(n min{k, log n log k}) time, while no such subquadratic-time solution for arbitrary k was known before. In contrast, if the embedding is not fixed, then we prove that it is the NP-complete to decide whether a planar graph admits a strict-orthogonal drawing with reflex face complexity k, for some k 2 O(1).