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Canonical Representations for Circular-Arc Graphs Using Flip Sets

Abstract

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.

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References

  1. 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. http://www.sciencedirect.com/science/article/pii/S0166218X15004308. https://doi.org/10.1016/j.dam.2015.08.023

  2. 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. http://drops.dagstuhl.de/opus/volltexte/2016/5727. https://doi.org/10.4230/LIPIcs.STACS.2016.26

  3. 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). http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/2298

  4. 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. 5.

    Gavril, F.: Algorithms on circular-arc graphs. Networks 4(4), 357–369 (1974). https://doi.org/10.1002/net.3230040407

  6. 6.

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

  7. 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). https://doi.org/10.1137/S0097539793260726

  8. 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). https://doi.org/10.1007/s00453-009-9304-5

  9. 9.

    Köbler, J., Kuhnert, S., Laubner, B., Verbitsky, O.: Interval graphs: canonical representations in logspace. SIAM J. Comput. 40(5), 1292–1315 (2011). https://doi.org/10.1137/10080395X

  10. 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). https://doi.org/10.1007/978-3-642-40313-2_56

  11. 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). http://www.sciencedirect.com/science/article/pii/S1570866716300016. https://doi.org/10.1016/j.jda.2016.03.001

  12. 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). http://www.sciencedirect.com/science/article/pii/S0012365X08002161. https://doi.org/10.1016/j.disc.2008.04.003

  13. 13.

    Lueker, G.S., Booth, K.S.: A linear time algorithm for deciding interval graph isomorphism. J. ACM 26(2), 183–195 (1979). https://doi.org/10.1145/322123.322125

  14. 14.

    McConnell, R.M.: Linear-time recognition of circular-arc graphs. Algorithmica 37(2), 93–147 (2003). https://doi.org/10.1007/s00453-003-1032-7

  15. 15.

    Stahl, F.W.: Circular genetic maps. J. Cell. Physiol. 70(S1), 1–12 (1967). https://doi.org/10.1002/jcp.1040700403

  16. 16.

    Tucker, A.: Characterizing circular-arc graphs. Bull. Am. Math. Soc., 76(6):1257–1260, 11 (1970). http://projecteuclid.org/euclid.bams/1183532398

  17. 17.

    Waterman, M.S., Griggs, J.R.: Interval graphs and maps of dna. Bull. Math. Biol. 48(2), 189–195 (1986). http://www.sciencedirect.com/science/article/pii/S0092824086800064. https://doi.org/10.1016/S0092-8240(86)80006-4

  18. 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)

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Acknowledgements

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

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Correspondence to Maurice Chandoo.

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Chandoo, M. Canonical Representations for Circular-Arc Graphs Using Flip Sets. Algorithmica 80, 3646–3672 (2018). https://doi.org/10.1007/s00453-018-0410-0

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Keywords

  • Canonization
  • Circular-arc graphs
  • Isomorphism

Mathematics Subject Classification

  • G.2.2 Graph Theory