Rectilinear Planarity of Partial 2-Trees

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