Document Open Access Logo

On the Independence Number of 1-Planar Graphs

Authors Therese Biedl, Prosenjit Bose, Babak Miraftab

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2024.13.pdf
  • Filesize: 0.8 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • 1-planar graph
  • independent set
  • minimum degree

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

An independent set in a graph is a set of vertices where no two vertices are adjacent to each other. A maximum independent set is the largest possible independent set that can be formed within a given graph G. The cardinality of this set is referred to as the independence number of G. This paper investigates the independence number of 1-planar graphs, a subclass of graphs defined by drawings in the Euclidean plane where each edge can have at most one crossing point. Borodin establishes a tight upper bound of six for the chromatic number of every 1-planar graph G, leading to a corresponding lower bound of n/6 for the independence number, where n is the number of vertices of G. In contrast, the upper bound for the independence number in 1-planar graphs is less studied. This paper addresses this gap by presenting upper bounds based on the minimum degree δ. A comprehensive table summarizes these upper bounds for various δ values, providing insights into achievable independence numbers under different conditions.

Cite As Get BibTex

Therese Biedl, Prosenjit Bose, and Babak Miraftab. On the Independence Number of 1-Planar Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SWAT.2024.13

Author Details

Therese Biedl
  • David R. Cheriton School of Computer Science, University of Waterloo, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Babak Miraftab
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

  • Biedl, Therese: Supported by NSERC; FRN RGPIN-2020-03958.
  • Bose, Prosenjit: Supported by NSERC.
  • Miraftab, Babak: Supported by NSERC.

References

  1. Michael O. Albertson. A lower bound for the independence number of a planar graph. Journal of Combinatorial Theory, Series B, 20(1):84-93, 1976. Google Scholar
  2. Vladimir E. Alekseev, Vadim V. Lozin, Dmitriy S. Malyshev, and Martin Milanic. The maximum independent set problem in planar graphs. In Proc. MFCS, volume 5162 of Lecture Notes in Computer Science, pages 96-107. Springer, 2008. Google Scholar
  3. Kenneth Appel and Wolfgang Haken. Every Planar Map is Four Colorable. American Mathematical Society, Providence, RI, 1989. Google Scholar
  4. Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. J. ACM, 41(1):153-180, 1994. Google Scholar
  5. Claude Berge. Graphes et hypergraphes. Dunod, Paris, 1970. Google Scholar
  6. T. Biedl and D. Wilkinson. Bounded-degree independent sets in planar graphs. Theory Comput. Syst., 38(3):253-278, 2005. Google Scholar
  7. T. Biedl and J. Wittnebel. Matchings in 1-planar graphs with large minimum degree. J. Graph Theory, 99(2):217-320, 2022. URL: https://doi.org/10.1002/jgt.22736.
  8. R. Bodendiek, H. Schumacher, and K. Wagner. Über 1-optimale graphen. Mathematische Nachrichten, 117(1):323-339, 1984. URL: https://doi.org/10.1002/mana.3211170125.
  9. J. Adrian Bondy and Uppaluri S. R. Murty. Graph Theory. Graduate Texts in Mathematics. Springer, 2008. Google Scholar
  10. O. V. Borodin. Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs. Metody Diskret. Analiz., 41:12-26, 108, 1984. Google Scholar
  11. Prosenjit Bose, Vida Dujmović, and David R. Wood. Induced subgraphs of bounded degree and bounded treewidth. In Graph-theoretic concepts in computer science, volume 3787 of Lecture Notes in Comput. Sci., pages 175-186. Springer, Berlin, 2005. URL: https://doi.org/10.1007/11604686_16.
  12. James E. Burns. The maximum independent set problem for cubic planar graphs. Networks, 19(3):373-378, 1989. Google Scholar
  13. Y. Caro and Y. Roditty. On the vertex-independence number and star decomposition of graphs. Ars Combin., 20:167-180, 1985. Google Scholar
  14. Zhi-Zhong Chen and Mitsuharu Kouno. A linear-time algorithm for 7-coloring 1-plane graphs. Algorithmica, 43(3):147-177, 2005. URL: https://doi.org/10.1007/S00453-004-1134-X.
  15. Norishige Chiba, Takao Nishizeki, and Nobuji Saito. An algorithm for finding a large independent set in planar graphs. Networks, 13(2):247-252, 1983. Google Scholar
  16. Marek Chrobak and Joseph Naor. An efficient parallel algorithm for computing a large independent set in planar graph. Algorithmica, 6(6):801-815, 1991. Google Scholar
  17. Daniel W. Cranston and Landon Rabern. Planar graphs have independence ratio at least 3/13. Electron. J. Comb., 23(3):3, 2016. Google Scholar
  18. Norm Dadoun and David G. Kirkpatrick. Parallel algorithms for fractional and maximal independent sets in planar graphs. Discret. Appl. Math., 27(1-2):69-83, 1990. Google Scholar
  19. David P. Dobkin and David G. Kirkpatrick. A linear algorithm for determining the separation of convex polyhedra. J. Algorithms, 6(3):381-392, 1985. Google Scholar
  20. Zdenek Dvorák and Matthias Mnich. Large independent sets in triangle-free planar graphs. SIAM J. Discret. Math., 31(2):1355-1373, 2017. Google Scholar
  21. M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some simplified NP-complete graph problems. Theor. Comput. Sci., 1(3):237-267, 1976. Google Scholar
  22. Xin He. A nearly optimal parallel algorithm for constructing maximal independent set in planar graphs. Theor. Comput. Sci., 61:33-47, 1988. Google Scholar
  23. D.V. Karpov. An upper bound on the number of edges in an almost planar bipartite graph. Journal of Mathematical Sciences, 196:737-746, 2014. URL: https://doi.org/10.1007/s10958-014-1690-9.
  24. David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM J. Comput., 12(1):28-35, 1983. Google Scholar
  25. Vadim V. Lozin and Martin Milanic. On the maximum independent set problem in subclasses of planar graphs. J. Graph Algorithms Appl., 14(2):269-286, 2010. Google Scholar
  26. C. E. Veni Madhavan. Approximation algorithm for maximum independent set in planar traingle-free [sic!] graphs. In Proc. FSTTCS, volume 181 of Lecture Notes in Computer Science, pages 381-392. Springer, 1984. Google Scholar
  27. Avner Magen and Mohammad Moharrami. Robust algorithms for on minor-free graphs based on the Sherali-Adams hierarchy. In Proc. APPROX, volume 5687 of Lecture Notes in Computer Science, pages 258-271. Springer, 2009. Google Scholar
  28. Bojan Mohar. Face covers and the genus problem for apex graphs. J. Comb. Theory, Ser. B, 82(1):102-117, 2001. Google Scholar
  29. Neil Robertson, Daniel P. Sanders, Paul D. Seymour, and Robin Thomas. The four-colour theorem. J. Comb. Theory, Ser. B, 70(1):2-44, 1997. Google Scholar
  30. Marcus Schaefer. The graph crossing number and its variants: a survey. Electron. J. Combin., DS21:90, 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact