Efficient Algorithms for Bend-minimum Orthogonal Drawings of Planar 3-Graphs
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Let G be a planar 3-graph (i.e., a planar graph with vertex degree at most three) with n vertices. We prove that a planar orthogonal drawing of G with the minimum number of bends can be computed in O(n^2) time. The minimum is taken over all planar embeddings of G. The most efficient known algorithm to solve this problem has complexity Õ(n^17/7), as proved by Chang and Yen for biconnected planar 3-graphs [SoCG Proc., 2017]. If either a distinguished edge or a distinguished vertex of G is constrained to be on the external face, a bend-minimum orthogonal drawing of G that respects the given constraint can be computed in O(n) time. An additional feature of our algorithms is that they compute drawings with at most two bends per edge, which is worst-case optimal.
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