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International Colloquium on Automata, Languages, and Programming

ICALP 2000: Automata, Languages and Programming pp 821–831Cite as

Computing the Girth of a Planar Graph

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1853))

Abstract

The girth of a graph G has been defined as the length of a shortest cycle of G. We design an O(n 5/4 log n) algorithm for finding the girth of an undirected n-vertex planar graph, giving the first o(n 2) algorithm for this problem. Our approach combines several techniques such as graph separation, hammock decomposition, covering of a planar graph with graphs of small tree-width, and dynamic shortest path computation. We discuss extensions and generalizations of our result.

This work was partially supported by the EPA grant R82-5207-01-0, EPSRC grant GR/M60750, and RTDF grant 98/99-0140.

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References

  1. Alon, Yuster and Zwick. Color-coding. Journal of the ACM, 42:844–856, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  2. Alon, Yuster and Zwick. Finding and counting given length cycles. Algorithmica, 17:209–223, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  3. Brenda S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(l):153–180, January 1994.

    Google Scholar 

  4. Bela Bollob⇒. Chromatic number, girth and maximal degree. Discrete Mathematics, 24:311–314, 1978.

    Article  MathSciNet  Google Scholar 

  5. R. J. Cook. Chromatic number and girth. Periodica Mathematica Hungarica, 6(l):103–107, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  6. Reinhard Diestel. Graph Theory. Springer-Verlag, Berlin, Heidelberg, New York, Tokio, 1997.

    MATH  Google Scholar 

  7. H. Djidjev, G. Pantziou, and C. Zaroliagis. Improved algorithms for dynamic shortest paths. Algorithmica, 1999. To appear.

    Google Scholar 

  8. Hristo N. Djidjev. A separator theorem. Compt. rend. Acad. bulg. Sci., 34:643–645, 1981.

    MATH  MathSciNet  Google Scholar 

  9. Hristo N. Djidjev. On the problem of partitioning planar graphs. SIAM Journal on Algebraic and Discrete Methods, 3:229–240, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  10. Hristo N. Djidjev. A linear algorithm for partitioning graphs of fixed genus. Serdica, 11:329–341, 1985.

    Google Scholar 

  11. David Eppstein. Subgraph isomorphism for planar graphs and related problems. In Proc. 6th Symp. Discrete Algorithms, pages 632–640. Assoc. Comput. Mach, and Soc. Industrial & Applied Math., 1995.

    Google Scholar 

  12. Paul Erdös. Graph theory and probability. Ganad. J. Math., 11:34–38, 1959.

    MATH  Google Scholar 

  13. G.N. Prederickson. Planar graph decomposition and all pairs shortest paths. Journal of the ACM, 38:162–204, 1991.

    Article  Google Scholar 

  14. John R. Gilbert, Joan P. Hutchinson, and Robert E. Tarjan. A separator theorem for graphs of bounded genus. J. Algorithms, 5:391–407, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Harary. Graph Theory. Addison-Wesley, Reading, Massachusetts, 1969.

    Google Scholar 

  16. A. Itai and M. Rodeh. Finding a minimum circuit in a graph. SIAM J. Computing, 7:413–423, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Klein, S. Rao, M. Rauch, and S. Subramanian. Faster shortest-path algorithms for planar graphs. In 26th ACM Symp. Theory of Computing, pages 27–37, 1994.

    Google Scholar 

  18. Richard J. Lipton and Robert E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math, 36:177–189, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  19. L. Lovasz. On chromatic number of finite set systems. Ada Math. Acad. Sci. Hun., 19:59–67, 1968.

    Article  MATH  MathSciNet  Google Scholar 

  20. Wolfgang Mader. Topological subgraphs in graphs of large girth. Gombinatorica, 18:405, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  21. B. Monien. The complexity of determining a shortest cycle of even length. Computing, 31:355–369, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  22. Carsten Thomassen. Paths, circuits and subdivisions. In Selected Topics in Graph Theory, ed. Lowell W. Beineke and Robin J. Wilson, Academic Press, volume 3. 1988.

    Google Scholar 

  23. Vazirani and Yannakakis. Pfafiian orientations, 0-1 permanents, and even cycles in directed graphs. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science, 25, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Warwick, Coventry, CV4 7AL, UK

    Hristo N. Djidjev

Authors
  1. Hristo N. Djidjev
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Editor information

Editors and Affiliations

  1. Department of Computer Sciences, University of Pisa, Corso Italia, 40, 56125, Pisa, Italy

    Ugo Montanari

  2. Center for Computer Sciences, University of Geneva, 24, Rue Général Dufour, 1211, Geneva 4, Switzerland

    José D. P. Rolim

  3. Department of Computer Sciences, ETH Zurich, 8092, Zurich, Switzerland

    Emo Welzl

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© 2000 Springer-Verlag Berlin Heidelberg

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Djidjev, H.N. (2000). Computing the Girth of a Planar Graph. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_69

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  • DOI: https://doi.org/10.1007/3-540-45022-X_69

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67715-4

  • Online ISBN: 978-3-540-45022-1

  • eBook Packages: Springer Book Archive

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