Skip to main content
Log in

Canonical Representations for Circular-Arc Graphs Using Flip Sets

  • Published:
Algorithmica Aims and scope Submit manuscript


We show that computing canonical representations for circular-arc (CA) graphs reduces to computing certain subsets of vertices called flip sets. For a broad class of CA graphs, which we call uniform, it suffices to compute a CA representation to find such flip sets. As a consequence canonical representations for uniform CA graphs can be obtained in polynomial-time. We then investigate what kind of CA graphs pose a challenge to this approach. This leads us to introduce the notion of restricted CA matrices and show that the canonical representation problem for CA graphs is logspace-reducible to that of restricted CA matrices. As a byproduct, we obtain the result that CA graphs without induced 4-cycles can be canonized in logspace.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others


  1. Cao, Y., Grippo, L.N., Safe, M.D.: Forbidden induced subgraphs of normal Helly circular-arc graphs: characterization and detection. Discrete Appl. Math., 216(Part 1):67–83 (2017). Special Graph Classes and Algorithms—in Honor of Professor Andreas Brandstädt on the Occasion of His 65th Birthday.

    Article  MathSciNet  Google Scholar 

  2. Chandoo, M.: Deciding circular-arc graph isomorphism in parameterized logspace. In: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), volume 47 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 26:1–26:13, Dagstuhl, Germany, 2016. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.

  3. Curtis, A., Lin, M., McConnell, R., Nussbaum, Y., Soulignac, F., Spinrad, J., Szwarcfiter, J.: Isomorphism of graph classes related to the circular-ones property. Discrete Math. Theor. Comput. Sci. (2013).

  4. Eschen, E.M.: Circular-Arc Graph Recognition and Related Problems. Ph.D. thesis, Vanderbilt University, Nashville, TN, USA, 1998. UMI Order No. GAX98-03921

  5. Gavril, F.: Algorithms on circular-arc graphs. Networks 4(4), 357–369 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  6. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, vol. 57). North-Holland Publishing Co., Amsterdam (2004)

    Chapter  Google Scholar 

  7. 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 

  8. Joeris, B.L., Lin, M.C., McConnell, R.M., Spinrad, J.P., Szwarcfiter, J.L.: Linear-time recognition of Helly circular-arc models and graphs. Algorithmica 59(2), 215–239 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  9. Köbler, J., Kuhnert, S., Laubner, B., Verbitsky, O.: Interval graphs: canonical representations in logspace. SIAM J. Comput. 40(5), 1292–1315 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. Köbler, J., Kuhnert, S., Verbitsky, O.: Helly circular-arc graph isomorphism is in logspace. In: Mathematical Foundations of Computer Science 2013, vol. 8087 of Lecture Notes in Computer Science, pp. 631–642. Springer, Berlin, Heidelberg (2013).

    Chapter  Google Scholar 

  11. Köbler, J., Kuhnert, S., Verbitsky, O. Solving the canonical representation and star system problems for proper circular-arc graphs in logspace. J. Discrete Algor. 38-41:38–49 (2016).

    Article  MathSciNet  Google Scholar 

  12. Lin, M.C., Szwarcfiter, J.L.: Characterizations and recognition of circular-arc graphs and subclasses: a survey. Discrete Math. 309(18):5618–5635 (2009). Combinatorics 2006, A Meeting in Celebration of Pavol Hell’s 60th Birthday (May 1–5, 2006).

    Article  MathSciNet  Google Scholar 

  13. Lueker, G.S., Booth, K.S.: A linear time algorithm for deciding interval graph isomorphism. J. ACM 26(2), 183–195 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  14. McConnell, R.M.: Linear-time recognition of circular-arc graphs. Algorithmica 37(2), 93–147 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  15. Stahl, F.W.: Circular genetic maps. J. Cell. Physiol. 70(S1), 1–12 (1967).

    Article  Google Scholar 

  16. Tucker, A.: Characterizing circular-arc graphs. Bull. Am. Math. Soc., 76(6):1257–1260, 11 (1970).

    Article  MathSciNet  Google Scholar 

  17. Waterman, M.S., Griggs, J.R.: Interval graphs and maps of dna. Bull. Math. Biol. 48(2), 189–195 (1986).

    Article  MathSciNet  Google Scholar 

  18. Wu, T.-H.: An \(O(n^3)\) Isomorphism Test for Circular-Arc Graphs. Ph.D. thesis, SUNY Stony Brook, New York, NY, USA (1983)

Download references


We thank the anonymous reviewers for their insightful comments and suggestions that helped us to improve the quality of this work.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Maurice Chandoo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chandoo, M. Canonical Representations for Circular-Arc Graphs Using Flip Sets. Algorithmica 80, 3646–3672 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification