Computing planar orthogonal drawings with the minimum number of bends is one of the most relevant topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number of bends, the number of vertices of degree at most two, and the treewidth of the input graph (Di Giacomo et al., 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by the number of vertices of degree at most two plus the number of bends. As a consequence, rectilinear planarity testing lies in \FPT~parameterized by the number of vertices of degree at most two.

翻译：计算具有最少弯曲数的平面正交图是图绘制中最重要的问题之一。已知该问题是NP难的，即使仅测试是否存在无弯曲的正交平面图（即直线平面图）也是如此（Garg 和 Tamassia，2001）。从参数化复杂度的角度来看，当以三个参数之和为参数时，该问题是固定参数可解的：弯曲数、度不超过2的顶点数以及输入图的树宽（Di Giacomo 等，2022）。我们通过证明当仅以度不超过2的顶点数加上弯曲数为参数时，该问题仍然是固定参数可解的，改进了这一结果。作为推论，直线平面性测试属于以度不超过2的顶点数为参数的\FPT类问题。