The Parametrized Complexity of the Segment Number

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.