Given a straight-line drawing of a graph, a {\em segment} is a maximal set of edges that form a line segment. Given a planar graph $G$, the {\em segment number} of $G$ is the minimum number of segments that can be achieved by any planar straight-line drawing of $G$. The {\em 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，一个“段”是构成一条线段的最大边集。给定一个平面图$G$，$G$的“段数”是$G$的所有平面直线绘制中可达到的最少段数。$G$的“线覆盖数”是支撑$G$的平面直线绘制中所有边的最少直线数。计算平面图的段数或线覆盖数是$\exists\mathbb{R}$-完全的，因此是NP难的。我们从参数化复杂度的角度研究段数的计算问题。我们证明该问题分别对于以下每个参数都是固定参数可处理的：顶点覆盖数、段数和线覆盖数。我们还考虑了段数和线覆盖数的着色变体。