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arXiv · 2308.15416

The Parametrized Complexity of the Segment Number

Sabine Cornelsen·Giordano Da Lozzo·Luca Grilli·Siddharth Gupta·Jan Kratochvíl·Alexander Wolff
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Abstract

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.

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Sabine Cornelsen, Giordano Da Lozzo, Luca Grilli, Siddharth Gupta, Jan Kratochvíl, Alexander Wolff. 2023-08-29. The Parametrized Complexity of the Segment Number. https://arxiv.org/abs/2308.15416

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