Given a straight-line drawing of a graph, a segment is a maximal set of edges that form a line segment. Given a planar graph $G$, the segment number of $G$ is the minimum number of segments that can be achieved by any planar straight-line drawing of $G$. The line cover number of $G$ is the minimum number of lines that support all the edges of a planar straight-line drawing of $G$. Computing the segment number or the line cover number of a planar graph is $\exists\mathbb{R}$-complete and, thus, NP-hard. We study the problem of computing the segment number from the perspective of parameterized complexity. We show that this problem is fixed-parameter tractable with respect to each of the following parameters: the vertex cover number, the segment number, and the line cover number. We also consider colored versions of the segment and the line cover number.

翻译：给定一个图的直线绘制，线段是指构成一条直线段的最大边集。对于平面图$G$，其线段数是指$G$的任何平面直线绘制所能达到的最小线段数量。$G$的线覆盖数是指支撑$G$的平面直线绘制中所有边所需的最小直线数量。计算平面图的线段数或线覆盖数是$\exists\mathbb{R}$-完全的，因此也是NP-难的。我们从参数化复杂度的角度研究计算线段数的问题。我们证明该问题对于以下每个参数都是固定参数可解的：顶点覆盖数、线段数以及线覆盖数。我们还考虑了线段数和线覆盖数的着色版本。