Skip to main content
SpringerLink
Log in

Circle Graph Isomorphism in Almost Linear Time

  • Conference paper
  • First Online:
Theory and Applications of Models of Computation (TAMC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13571))

Abstract

Circle graphs are intersection graphs of chords of a circle. In this paper, we present a new algorithm for the circle graph isomorphism problem running in time \({\mathcal {O}}((n+m)\alpha (n+m))\) where n is the number of vertices, m is the number of edges and \(\alpha \) is the inverse Ackermann function. Our algorithm is based on the minimal split decomposition [Cunnigham, 1982] and uses the state-of-art circle graph recognition algorithm [Gioan, Paul, Tedder, Corneil, 2014] in the same running time. It improves the running time \({\mathcal {O}}(nm)\) of the previous algorithm [Hsu, 1995] based on a similar approach.

P. Zeman—Was supported by the Swiss National Science Foundation project PP00P2-202667. While at Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Peter Zeman was supported by GAČR 20-15576S.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The design and analysis of computer algorithms. Addison-Wesley Publishing Company (1974)

    Google Scholar 

  2. Booth, K.S.: Lexicographically least circular substrings. Inf. Process. Lett. 10(4–5), 240–242 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bouchet, A.: Reducing prime graphs and recognizing circle graphs. Combinatorica 7(3), 243–254 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchet, A.: Unimodularity and circle graphs. Discret. Math. 66(1–2), 203–208 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cunningham, W.H.: Decomposition of directed graphs. SIAM J. Algebraic Discrete Methods 3(2), 214–228 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cunningham, W.H., Edmonds, J.: A combinatorial decomposition theory. Can. J. Math. 32(3), 734–765 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Curtis, A.R., et al.: Isomorphism of graph classes related to the circular-ones property. Discrete Math. Theor. Comput. Sci. 15(1), 157–182 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Dahlhaus, E.: Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition. J. Algorithms 36(2), 205–240 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Even, S., Itai, A.: Queues, stacks, and graphs. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, pp. 71–76 (1971)

    Google Scholar 

  10. de Fraysseix, H.: Local complementation and interlacement graphs. Discret. Math. 33(1), 29–35 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. de Fraysseix, H., de Mendez, P.O.: On a characterization of gauss codes. Discrete Comput. Geom. 22(2), 287–295 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gabor, C.P., Supowit, K.J., Hsu, W.: Recognizing circle graphs in polynomial time. J. ACM 36(3), 435–473 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient circle graph recognition. Algorithmica 69(4), 759–788 (2014). https://doi.org/10.1007/s00453-013-9745-8

    Article  MathSciNet  MATH  Google Scholar 

  14. Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient split decomposition via graph-labelled trees. Algorithmica 69(4), 789–843 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hsu, W.L.: \(O(M \cdot N)\) algorithms for the recognition and isomorphism problems on circular-arc graphs. SIAM J. Comput. 24(3), 411–439 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, M.C., Soulignac, F.J., Szwarcfiter, J.L.: A simple linear time algorithm for the isomorphism problem on proper circular-arc graphs. In: Gudmundsson, J. (ed.) SWAT 2008. LNCS, vol. 5124, pp. 355–366. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69903-3_32

    Chapter  MATH  Google Scholar 

  17. Naji, W.: Graphes de cordes: une caracterisation et ses applications, Ph. D. thesis, l’Université Scientifique et Médicale de Grenoble (1985)

    Google Scholar 

  18. Oum, S.: Rank-width and vertex-minors. J. Comb. Theory Ser. B 95(1), 79–100 (2005)

    Google Scholar 

  19. Shiloach, Y.: Fast canonization of circular strings. J. Algorithms 2(2), 107–121 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Spinrad, J.P.: Recognition of circle graphs. J. Algorithms 16(2), 264–282 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Spinrad, J.P.: Efficient graph representations. Field Institute Monographs (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. OrgPad, Prague, Czech Republic

    Vít Kalisz & Pavel Klavík

  2. Institut de Mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland

    Peter Zeman

Authors
  1. Vít Kalisz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavel Klavík
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Zeman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Zeman .

Editor information

Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. University of New Brunswick, Fredericton, NB, Canada

    Donglei Du

  3. Tianjin University of Technology, Tianjin, China

    Chenchen Wu

  4. Beijing University of Technology, Beijing, China

    Dachuan Xu

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kalisz, V., Klavík, P., Zeman, P. (2022). Circle Graph Isomorphism in Almost Linear Time. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Theory and Applications of Models of Computation. TAMC 2022. Lecture Notes in Computer Science, vol 13571. Springer, Cham. https://doi.org/10.1007/978-3-031-20350-3_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-20350-3_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-20349-7

  • Online ISBN: 978-3-031-20350-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation