A graph is rectilinear planar if it admits a planar orthogonal drawing without bends. While testing rectilinear planarity is NP-hard in general (Garg and Tamassia, 2001), 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)-time algorithm to test rectilinear planarity of partial 2-trees, which improves over the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover, for partial 2-trees where no two parallel-components in a biconnected component 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 several years ago (Di Battista et al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up in an orthogonal drawing of the graph.

翻译：图若存在无弯曲的平面正交绘制，则称为直线平面图。虽然直线平面性测试在一般情况下是NP困难的（Garg和Tamassia，2001年），但在部分2树（即双连通分量均为串并联结构的图）上确立其复杂度的严格上界仍是一个长期未解问题。我们提出了一种新的O(n²)时间算法，用于测试部分2树的直线平面性，该算法改进了当前最优的O(n³ log n)复杂度界（Di Giacomo等，2022年）。此外，对于双连通分量中任意两个平行分量不共享极点的部分2树，我们能够实现最优的O(n)时间复杂性。我们的算法基于对正交螺旋性概念的深入研究和理解——该概念由Di Battista等人（1998年）提出，用于描述子图的正交绘制在整体图的正交绘制中的卷曲程度。