Rectilinear Planarity of Partial 2-Trees

A graph is rectilinear planar if it admits a orthogonal drawing without bends. While testing planarity NP-hard in general, long-standing open problem to establish tight upper bound on its complexity for partial 2-trees, i.e., graphs whose biconnected components are series-parallel. We describe new $$O(n^2 \log ^2 n)$$ -time algorithm test of which improves over the current best $$O(n^3 . Moreover, series-parallel where no two parallel-components share pole, we able achieve optimal O(n)-time complexity. Our algorithms based an extensive study and deeper understanding notion spirality, introduced 1998 how much subgraph rolled-up graph.