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A Simpler Linear-Time Recognition of Circular-Arc Graphs

  • Haim Kaplan
  • Yahav Nussbaum
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)

Abstract

We give a linear time recognition algorithm for circular-arc graphs. Our algorithm is much simpler than the linear time recognition algorithm of McConnell [10] (which is the only linear time recognition algorithm previously known). Our algorithm is a new and careful implementation of the algorithm of Eschen and Spinrad [4, 5]. We also tighten the analysis of Eschen and Spinrad.

Keywords

Linear Time Interval Graph Combinatorial Auction Cyclic Order Left Endpoint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Conitzer, V., Derryberry, J., Sandholm, T.: Combinatorial auctions with structured item graphs. In: Proceedings of the Nineteenth National Conference on Artificial Intelligence, pp. 212–218 (2004)Google Scholar
  3. 3.
    Deng, X., Hell, P., Huang, J.: Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs. SIAM J. Comput. 25(2), 390–403 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Eschen, E.M.: Circular-arc graph recognition and related problems. PhD thesis, Department of Computer Science, Vanderbilt University (1997)Google Scholar
  5. 5.
    Eschen, E.M., Spinrad, J.P.: An O(n 2) algorithm for circular-arc graph recognition. In: SODA 1993: Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 128–137 (1993)Google Scholar
  6. 6.
    Hsu, W.-L.: O(mn) algorithms for the recognition and isomorphism problems on circular-arc graphs. SIAM J. Comput. 24(3), 411–439 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hsu, W.-L., McConnell, R.M.: PC-trees and circular-ones arrangements. Theor. Comput. Sci. 296(1), 99–116 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lin, M.C., Szwarcfiter, J.L.: Efficient construction of unit circular-arc models. In: SODA 2006: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 309–315 (2006)Google Scholar
  9. 9.
    Ma, T.-H., Spinrad, J.P.: Avoiding matrix multiplication. In: Möhring, R.H. (ed.) WG 1990. LNCS, vol. 484, pp. 61–71. Springer, Heidelberg (1991)Google Scholar
  10. 10.
    McConnell, R.M.: Linear-time recognition of circular-arc graphs. Algorithmica 37(2), 93–147 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Mathematics 201(1-3), 189–241 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Spinrad, J.P.: Circular-arc graphs with clique cover number two. Journal of Combinatorial Theory Series B 44(3), 300–306 (1988)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Spinrad, J.P.: Efficient Graph Representations. Fields Institute Monographs, vol. 19. American Mathematical Society (2003)Google Scholar
  14. 14.
    Spinrad, J.P., Valdes, J.: Recognition and isomorphism of two dimensional partial orders. In: Díaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 676–686. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  15. 15.
    Stefanakos, S., Erlebach, T.: Routing in all-optical ring networks revisited. In: Proceedings of the 9th IEEE Symposium on Computers and Communication, pp. 288–293 (2004)Google Scholar
  16. 16.
    Tucker, A.C.: An efficient test for circular-arc graphs. SIAM J. Comput. 9(1), 1–24 (1980)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Haim Kaplan
    • 1
  • Yahav Nussbaum
    • 1
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael

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