Document Open Access Logo

Linear-Time Recognition of Map Graphs with Outerplanar Witness

Authors Matthias Mnich, Ignaz Rutter, Jens M. Schmidt

PDF
Thumbnail PDF

File

LIPIcs.SWAT.2016.5.pdf
  • Filesize: 0.51 MB
  • 14 pages

Document Identifiers

Author Details

Matthias Mnich
Ignaz Rutter
Jens M. Schmidt

Cite As Get BibTex

Matthias Mnich, Ignaz Rutter, and Jens M. Schmidt. Linear-Time Recognition of Map Graphs with Outerplanar Witness. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SWAT.2016.5

Abstract

Map graphs generalize planar graphs and were introduced by Chen, Grigni and Papadimitriou [STOC 1998, J.ACM 2002]. They showed that the problem of recognizing map graphs is in NP by proving the existence of a planar witness graph W. Shortly after, Thorup [FOCS 1998] published a polynomial-time recognition algorithm for map graphs. However, the run time of this algorithm is estimated to be Omega(n^{120}) for n-vertex graphs, and a full description of its details remains unpublished.
  
We give a new and purely combinatorial algorithm that decides whether a graph G is a map graph having an outerplanar witness W. This is a step towards a first combinatorial recognition algorithm for general map graphs. The algorithm runs in time and space O(n+m). In contrast to Thorup's approach, it computes the witness graph W in the affirmative case.

Subject Classification

Keywords
  • Algorithms and data structures
  • map graphs
  • recognition
  • planar graphs

Metrics

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

References

  1. F. J. Brandenburg. On 4-map graphs and 1-planar graphs and their recognition problem. Technical report, ArXiv, 2015. URL: http://arxiv.org/abs/1509.03447.
  2. Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou. Map graphs. J. ACM, 49(2):127-138, 2002. Google Scholar
  3. Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou. Recognizing hole-free 4-map graphs in cubic time. Algorithmica, 45(2):227-262, 2006. Google Scholar
  4. Z.-Z. Chen, X. He, and M.-Y. Kao. Nonplanar topological inference and political-map graphs. In Proc. SODA 1999, pages 195-204, 1999. Google Scholar
  5. E. D. Demaine, F. V. Fomin, M. Hajiaghayi, and D. M. Thilikos. Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs. ACM Trans. Algorithms, 1(1):33-47, 2005. Google Scholar
  6. G. Di Battista and R. Tamassia. On-Line Maintenance of Triconnected Components with SPQR-Trees. Algorithmica, 15(4):302-318, 1996. Google Scholar
  7. G. Di Battista and R. Tamassia. On-Line Planarity Testing. SIAM J. Comput., 25(5):956-997, 1996. Google Scholar
  8. F. V. Fomin, D. Lokshtanov, N. Misra, and S. Saurabh. Planar ℱ-deletion: Approximation, kernelization and optimal FPT algorithms. Proc. FOCS 2012, pages 470-479, 2012. Google Scholar
  9. F. V. Fomin, D. Lokshtanov, and S. Saurabh. Bidimensionality and geometric graphs. In Proc. SODA 2012, pages 1563-1575, 2012. Google Scholar
  10. A. Grigoriev and H. L. Bodlaender. Algorithms for graphs embeddable with few crossings per edge. Algorithmica, 49(1):1-11, 2007. URL: http://dx.doi.org/10.1007/s00453-007-0010-x.
  11. C. Gutwenger and P. Mutzel. A linear time implementation of SPQR-trees. In Proc. GD 2000, volume 1984 of Lecture Notes Comput. Sci., pages 77-90, 2001. Google Scholar
  12. J. Hopcroft and R. Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, 1974. Google Scholar
  13. J. E. Hopcroft and R. E. Tarjan. Dividing a graph into triconnected components. SIAM J. Comput., 2(3):135-158, 1973. Google Scholar
  14. V. P. Korzhik and B. Mohar. Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory, 72(1):30-71, 2013. Google Scholar
  15. Y. L. Lin and S. S. Skiena. Algorithms for square roots of graphs. SIAM J. Discrete Math., 8(1):99-118, 1995. Google Scholar
  16. R. M. McConnell, K. Mehlhorn, S. Näher, and P. Schweitzer. Certifying algorithms. Comput. Sci. Review, 5(2):119-161, 2011. Google Scholar
  17. M. M. Sysło. Characterizations of outerplanar graphs. Discrete Math., 26(1):47-53, 1979. Google Scholar
  18. M. Thorup. Map graphs in polynomial time. In Proc. FOCS 1998, pages 396-405, 1998. Google Scholar
  19. K. Wagner. Über eine Eigenschaft der ebenen Komplexe. Math. Ann., 114(1):570-590, 1937. 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