LIPIcs.GD.2024.42.pdf
- Filesize: 0.5 MB
- 3 pages
Document
From Planar via Outerplanar to Outerpath – Engineering NP-Hardness Constructions (Poster Abstract)
Authors
Johannes Zink 
-
Part of:
Volume:
32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Graph Drawing and Network Visualization (GD) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-10-28
File
Document Identifiers
Author Details
Cite As Get BibTex
Joshua Geis and Johannes Zink. From Planar via Outerplanar to Outerpath – Engineering NP-Hardness Constructions (Poster Abstract). In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 42:1-42:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.42
Abstract
A typical question in graph drawing is to determine, for a given graph drawing style, the boundary between polynomial-time solvability and NP-hardness. For two examples from the area of drawing graphs with few slopes, we sharpen this boundary. We suggest a framework for a certain type of NP-hardness constructions where graphs have some parts that can only be realized as rigid structures, whereas other parts allow a controllable degree of flexibility. Starting with an NP-complete problem involving planarity (here, we use planar monotone rectilinear 3-SAT), we consider first a reduction to a planar graph, which can be adjusted to an outerplanar graph, and finally to an outerpath. An outerplanar graph is a graph admitting an outerplanar drawing, that is, a crossing-free drawing where every vertex lies on the outer face, and an outerpath is a graph admitting an outerplanar drawing where the weak dual is a path. The (weak) dual of a graph drawing is the graph that has a node for every (inner) face and a link if two faces share an edge. Specifically, we first show that, for every upward-planar directed outerpath G, it is NP-hard to decide whether G admits an upward-planar straight-line drawing where every edge has one of three distinct slopes, and we second show that, for every undirected outerpath G, it is NP-hard to decide whether G admits a proper level-planar straight-line drawing where every edge has one of two distinct slopes. For both problems, NP-hardness has been known before for outerplanar graphs.
Subject Classification
ACM Subject Classification
- Mathematics of computing → Graph algorithms
Keywords
- NP-hardness
- outerplanar
- outerpath
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Paola Bertolazzi, Giuseppe Di Battista, Giuseppe Liotta, and Carlo Mannino. Upward drawings of triconnected digraphs. Algorithmica, 12(6):476-497, 1994. URL: https://doi.org/10.1007/BF01188716.
- Mark de Berg and Amirali Khosravi. Optimal binary space partitions for segments in the plane. International Journal of Computational Geometry and Applications, 22(3):187-206, 2012. URL: https://doi.org/10.1142/s0218195912500045.
- Giuseppe Di Battista and Enrico Nardelli. Hierarchies and planarity theory. IEEE Transactions on Systems, Man, and Cybernetics, 18(6):1035-1046, 1988. URL: https://doi.org/10.1109/21.23105.
- Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing, 31(2):601-625, 2001. URL: https://doi.org/10.1137/S0097539794277123.
- Joshua Geis. Aufwärtsplanare Zeichnungen mit drei Steigungen von Außenpfaden. Bachelor’s thesis, Institut für Informatik, Universität Würzburg, 2022. (in German). URL: https://www1.pub.informatik.uni-wuerzburg.de/pub/theses/2022-geis-bachelor.pdf.
- Lenwood S. Heath and Sriram V. Pemmaraju. Recognizing leveled-planar dags in linear time. In Proc. Symposium on Graph Drawing (GD'95), pages 300-311. Springer, 1995. URL: https://doi.org/10.1007/BFb0021813.
- Udo Hoffmann. On the complexity of the planar slope number problem. Journal of Graph Algorithms and Applications, 21(2):183-193, 2017. URL: https://doi.org/10.7155/jgaa.00411.
- Michael Jünger, Sebastian Leipert, and Petra Mutzel. Level planarity testing in linear time. In Proc. 6th International Symposium on Graph Drawing (GD'98), pages 224-237. Springer, 1998. URL: https://doi.org/10.1007/3-540-37623-2_17.
- Jonathan Klawitter and Johannes Zink. Upward planar drawings with three and more slopes. In Proc. 29th International Symposium on Graph Drawing & Network Visualization (GD'21), pages 149-165. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92931-2_11.
- Jonathan Klawitter and Johannes Zink. Upward planar drawings with three and more slopes. Journal of Graph Algorithms and Applications, 27(2):49-70, 2023. URL: https://doi.org/10.7155/jgaa.00617.
-
Rebecca Kraus. Level-außenplanare Zeichnungen mit wenigen Steigungen. Bachelor’s thesis, Institut für Informatik, Universität Würzburg, 2020. (in German).
- Martin Nöllenburg. Automated drawing of metro maps. Diploma thesis, University of Karlsruhe (TH), 2005. URL: https://i11www.iti.kit.edu/extra/publications/n-admm-05da.pdf.
- Achilleas Papakostas. Upward planarity testing of outerplanar DAGs. In Proc. DIMACS International Workshop on Graph Drawing (GD'94), pages 298-306, 1994. URL: https://doi.org/10.1007/3-540-58950-3_385.
- Valentin Quapil. Upward and upward-planar drawings with limited slopes. Bachelor’s Thesis, Karlsruhe Institute of Technology, 2021. URL: https://i11www.iti.kit.edu/_media/teaching/theses/ba-quapil-21.pdf.
- Johannes Zink. Algorithms for Drawing Graphs and Polylines with Straight-Line Segments. PhD thesis, Universität Würzburg, 2024. URL: https://doi.org/10.25972/OPUS-35475.