OrthogonalPlanarity is decidable in polynomial time for bounded-treewidth planar graphs, and in O(n^3 log n) time for series-parallel graphs.
Algorithmica 68(4), 859–885 (2014)
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Sketched Representations and Orthogonal Planarity of Bounded Treewidth Graphs
OrthogonalPlanarity is decidable in polynomial time for bounded-treewidth planar graphs, and in O(n^3 log n) time for series-parallel graphs.