eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-07-02
99:1
99:17
10.4230/LIPIcs.ICALP.2024.99
article
Constrained Level Planarity Is FPT with Respect to the Vertex Cover Number
Klemz, Boris
1
https://orcid.org/0000-0002-4532-3765
Sieper, Marie Diana
1
https://orcid.org/0009-0003-7491-2811
Universität Würzburg, Germany
The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity, each level y is equipped with a partial order ≺_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of ≺_y. Constrained Level Planarity is known to be a remarkably difficult problem: previous results by Klemz and Rote [ACM Trans. Alg.'19] and by Brückner and Rutter [SODA'17] imply that it remains NP-hard even when restricted to graphs whose tree-depth and feedback vertex set number are bounded by a constant and even when the instances are additionally required to be either proper, meaning that each edge spans two consecutive levels, or ordered, meaning that all given partial orders are total orders. In particular, these results rule out the existence of FPT-time (even XP-time) algorithms with respect to these and related graph parameters (unless P=NP). However, the parameterized complexity of Constrained Level Planarity with respect to the vertex cover number of the input graph remained open.
In this paper, we show that Constrained Level Planarity can be solved in FPT-time when parameterized by the vertex cover number. In view of the previous intractability statements, our result is best-possible in several regards: a speed-up to polynomial time or a generalization to the aforementioned smaller graph parameters is not possible, even if restricting to proper or ordered instances.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol297-icalp2024/LIPIcs.ICALP.2024.99/LIPIcs.ICALP.2024.99.pdf
Parameterized Complexity
Graph Drawing
Planar Poset Diagram
Level Planarity
Constrained Level Planarity
Vertex Cover
FPT
Computational Geometry