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On Computing the Vertex Connectivity of 1-Plane Graphs

Authors Therese Biedl , Karthik Murali

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Author Details

Therese Biedl
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Karthik Murali
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

  • Biedl, Therese: Supported by NSERC.
  • Murali, Karthik: Research done in part for the author’s Master’s thesis at University of Waterloo [Murali, 2022].

Cite AsGet BibTex

Therese Biedl and Karthik Murali. On Computing the Vertex Connectivity of 1-Plane Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.23

Abstract

A graph is called 1-plane if it has an embedding in the plane where each edge is crossed at most once by another edge. A crossing of a 1-plane graph is called an ×-crossing if there are no other edges connecting the endpoints of the crossing (apart from the crossing pair of edges). In this paper, we show how to compute the vertex connectivity of a 1-plane graph G without ×-crossings in linear time. To do so, we show that for any two vertices u,v in a minimum separating set S, the distance between u and v in an auxiliary graph Λ(G) (obtained by planarizing G and then inserting into each face a new vertex adjacent to all vertices of the face) is small. It hence suffices to search for a minimum separating set in various subgraphs Λ_i of Λ(G) with small diameter. Since Λ(G) is planar, the subgraphs Λ_i have small treewidth. Each minimum separating set S then gives rise to a partition of Λ_i into three vertex sets with special properties; such a partition can be found via Courcelle’s theorem in linear time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • 1-Planar Graph
  • Connectivity
  • Linear Time
  • Treewidth

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References

  1. Christopher Auer, Franz J Brandenburg, Andreas Gleißner, and Josef Reislhuber. 1-planarity of graphs with a rotation system. Journal of Graph Algorithms and Applocations, 19(1):67-86, 2015. URL: https://doi.org/10.7155/jgaa.00347.
  2. Therese Biedl and Karthik Murali. On computing the vertex connectivity of 1-plane graphs. CoRR, abs/2212.06782, 2022. URL: https://doi.org/10.48550/arXiv.2212.06782.
  3. Rainer Bodendiek, Heinz Schumacher, and Klaus Wagner. Bemerkungen zu einem Sechsfarbenproblem von G Ringel. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 53:41-52, 1983. URL: https://doi.org/10.1007/BF02941309.
  4. Hans Bodlaender and Arie Koster. Combinatorial optimization on graphs of bounded treewidth. The Computer Journal, 51(3):255-269, 2008. URL: https://doi.org/10.1093/comjnl/bxm037.
  5. Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM Journal on Computing, 42(5):1803-1829, 2013. URL: https://doi.org/10.1137/120872310.
  6. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  7. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  8. David Eppstein. Subgraph isomorphism in planar graphs and related problems. Journal of Graph Algorithms and Applications, 3(3):1-27, 1999. URL: https://doi.org/10.7155/jgaa.00014.
  9. Igor Fabrici, Jochen Harant, Tomás Madaras, Samuel Mohr, Roman Soták, and Carol T Zamfirescu. Long cycles and spanning subgraphs of locally maximal 1-planar graphs. Journal of Graph Theory, 95(1):125-137, 2020. URL: https://doi.org/10.1002/jgt.22542.
  10. Fedor V Fomin and Dimitrios M Thilikos. New upper bounds on the decomposability of planar graphs. Journal of Graph Theory, 51(1):53-81, 2006. URL: https://doi.org/10.1002/jgt.20121.
  11. Sebastian Forster, Danupon Nanongkai, Liu Yang, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Computing and testing small connectivity in near-linear time and queries via fast local cut algorithms. In ACM-SIAM Symposium on Discrete Algorithms, pages 2046-2065. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.126.
  12. Harold N Gabow. Using expander graphs to find vertex connectivity. Journal of the ACM, 53(5):800-844, 2006. URL: https://doi.org/10.1145/1183907.1183912.
  13. Yu Gao, Jason Li, Danupon Nanongkai, Richard Peng, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Deterministic graph cuts in subquadratic time: Sparse, balanced, and k-vertex. CoRR, abs/1910.07950, 2019. URL: https://arxiv.org/abs/1910.07950.
  14. Alexander Grigoriev and Hans L Bodlaender. Algorithms for graphs embeddable with few crossings per edge. Algorithmica, 49(1):1-11, 2007. URL: https://doi.org/10.1007/s00453-007-0010-x.
  15. Carsten Gutwenger and Petra Mutzel. A linear time implementation of SPQR-trees. In Graph Drawing, volume 1984 of Lecture Notes in Computer Science, pages 77-90. Springer, 2000. URL: https://doi.org/10.1007/3-540-44541-2_8.
  16. Monika R Henzinger, Satish Rao, and Harold N Gabow. Computing vertex connectivity: new bounds from old techniques. Journal of Algorithms, 34(2):222-250, 2000. URL: https://doi.org/10.1006/jagm.1999.1055.
  17. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs, Communications of NII Shonan Meetings. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  18. John E Hopcroft and Robert Endre Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  19. Arkady Kanevsky and Vijaya Ramachandran. Improved algorithms for graph four-connectivity. Journal of Computer and System Sciences, 42(3):288-306, 1991. URL: https://doi.org/10.1016/0022-0000(91)90004-O.
  20. Daniel Kleitman. Methods for investigating connectivity of large graphs. IEEE Transactions on Circuit Theory, 16(2):232-233, 1969. URL: https://doi.org/10.1109/TCT.1969.1082941.
  21. Stephen Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. An annotated bibliography on 1-planarity. Computer Science Review, 25:49-67, 2017. URL: https://doi.org/10.1016/j.cosrev.2017.06.002.
  22. Vladimir P Korzhik and Bojan Mohar. Minimal obstructions for 1-immersions and hardness of 1-planarity testing. Journal of Graph Theory, 72(1):30-71, 2013. URL: https://doi.org/10.1002/jgt.21630.
  23. Jean-Paul Laumond. Connectivity of plane triangulations. Information Processing Letters, 34(2):87-96, 1990. URL: https://doi.org/10.1016/0020-0190(90)90142-K.
  24. Jason Li, Danupon Nanongkai, Debmalya Panigrahi, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai. Vertex connectivity in poly-logarithmic max-flows. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 317-329, 2021. URL: https://doi.org/10.1145/3406325.3451088.
  25. Nathan Linial, László Lovász, and Avi Wigderson. Rubber bands, convex embeddings and graph connectivity. Combinatorica, 8(1):91-102, 1988. URL: https://doi.org/10.1007/BF02122557.
  26. Karthik Murali. Testing vertex connectivity of bowtie 1-plane graphs. Master’s thesis, David R. Cheriton School of Computer Science, University of Waterloo, 2022. Available at URL: https://uwspace.uwaterloo.ca/.
  27. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. URL: https://doi.org/10.1007/BF01758778.
  28. Gerhard Ringel. Ein Sechsfarbenproblem auf der Kugel. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 29(1):107-117, 1965. URL: https://doi.org/10.1007/BF02996313.
  29. Robert E Tarjan. Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2):146-160, 1972. URL: https://doi.org/10.1137/0201010.
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