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Computer Science > Computational Geometry

arXiv:2308.15416 (cs)
[Submitted on 29 Aug 2023 (v1), last revised 2 Jul 2024 (this version, v3)]

Title:The Parametrized Complexity of the Segment Number

Authors: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.
Comments: The conference version of this paper appeared in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023)
Subjects: Computational Geometry (cs.CG)
Cite as: arXiv:2308.15416 [cs.CG]
  (or arXiv:2308.15416v3 [cs.CG] for this version)
  https://doi.org/10.48550/arXiv.2308.15416

Submission history

From: Alexander Wolff [view email]
[v1] Tue, 29 Aug 2023 16:21:08 UTC (271 KB)
[v2] Fri, 1 Sep 2023 06:35:15 UTC (271 KB)
[v3] Tue, 2 Jul 2024 15:50:42 UTC (271 KB)
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