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：计算具有最少弯折数的平面正交图是图绘制中最相关的话题之一。即使我们只想测试是否存在无弯折的正交图（即直线型正交图），该问题已知为NP难问题（Garg 和 Tamassia，2001）。从参数化复杂度的角度看，当参数化为三个参数之和时，该问题是固定参数可处理的：弯折数、度数不超过二的顶点数以及输入图的树宽（Di Giacomo 等人，2022）。我们改进了这一结果，表明该问题在仅以度数不超过二的顶点数加上弯折数为参数时，仍然是固定参数可处理的。因此，直线型正交图可测试性属于以度数不超过二的顶点数为参数的 \FPT 类。