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Testing Isomorphism of Chordal Graphs of Bounded Leafage is Fixed-Parameter Tractable (Extended Abstract)

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Graph-Theoretic Concepts in Computer Science (WG 2022)

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Abstract

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter.

The full version of this paper is available on arXiv [2]. Roman Nedela was supported by GAČR 20-15576S and APVV-19-0308. Peter 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.

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Notes

  1. 1.

    After posting our paper on the arXiv [2], we found a paper [7] containing an FPT algorithm testing isomorphism of T-graphs for every fixed tree T. This gives an alternative FPT algorithm for chordal graphs of leafage \(\ell \).

  2. 2.

    Note that the composition \(fGf^{-1}\) is defined from left to right.

References

  1. Agaoglu, D., Hlinený, P.: Isomorphism problem for \(S_d\)-graphs. In: Esparza, J., Král’, D. (eds.) 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, Prague, Czech Republic, 24–28 August 2020. LIPIcs, vol. 170, pp. 4:1–4:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.MFCS.2020.4

  2. Arvind, V., Nedela, R., Ponomarenko, I., Zeman, P.: Testing isomorphism of chordal graphs of bounded leafage is fixed-parameter tractable (2021). https://arxiv.org/abs/2107.10689

  3. Arvind, V., Das, B., Köbler, J., Toda, S.: Colored hypergraph isomorphism is fixed parameter tractable. Algorithmica 71(1), 120–138 (2015). https://doi.org/10.1007/s00453-013-9787-y

  4. Babai, L.: Graph isomorphism in quasipolynomial time [extended abstract]. In: Wichs, D., Mansour, Y. (eds.) Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 684–697. ACM (2016). https://doi.org/10.1145/2897518.2897542

  5. Biro, M., Hujter, M., Tuza, Z.: Precoloring extension I. Interval graphs. Discret. Math. 100(1–3), 267–279 (1992)

    Article  MathSciNet  Google Scholar 

  6. Brand, N.: Isomorphisms of cyclic combinatorial objects. Discret. Math. 78(1-2), 73–81 (1989). https://doi.org/10.1016/0012-365X(89)90162-3

  7. Çagirici, D.A., Hlinený, P.: Isomorphism testing for T-graphs in FPT. In: Mutzel, P., Rahman, M.S., Slamin (eds.) WALCOM 2022. LNCS, vol. 13174, pp. 239–250. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-96731-4_20

    Chapter  Google Scholar 

  8. Chaplick, S., Töpfer, M., Voborník, J., Zeman, P.: On H-topological intersection graphs. In: Bodlaender, H.L., Woeginger, G.J. (eds.) WG 2017. LNCS, vol. 10520, pp. 167–179. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68705-6_13

    Chapter  Google Scholar 

  9. Chaplick, S., Zeman, P.: Combinatorial problems on H-graphs. Electron. Notes Discret. Math. 61, 223–229 (2017). https://doi.org/10.1016/j.endm.2017.06.042

  10. Chaplick, S., Zeman, P.: Isomorphism-completeness for H-graphs (2021). https://kam.mff.cuni.cz/pizet/gic.pdf

  11. Chen, G., Ponomarenko, I.: Coherent Configurations. Central China Normal University Press, Wuhan (2019). http://www.pdmi.ras.ru/~inp/ccNOTES.pdf

  12. Fomin, F.V., Golovach, P.A., Raymond, J.-F.: On the tractability of optimization problems on H-graphs. Algorithmica 82(9), 2432–2473 (2020). https://doi.org/10.1007/s00453-020-00692-9

    Article  MathSciNet  MATH  Google Scholar 

  13. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory Series B 16(1), 47–56 (1974). https://doi.org/10.1016/0095-8956(74)90094-X. https://www.sciencedirect.com/science/article/pii/009589567490094X

  14. Grohe, M., Neuen, D., Schweitzer, P.: A faster isomorphism test for graphs of small degree. In: 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, 7–9 October 2018, pp. 89–100. IEEE Computer Society (2018). https://doi.org/10.1109/FOCS.2018.00018

  15. Grohe, M., Wiebking, D., Neuen, D.: Isomorphism testing for graphs excluding small minors. In: 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, 16–19 November 2020, pp. 625–636. IEEE (2020). https://doi.org/10.1109/FOCS46700.2020.00064

  16. Habib, M., Stacho, J.: Polynomial-time algorithm for the leafage of chordal graphs. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 290–300. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04128-0_27

    Chapter  Google Scholar 

  17. Lin, I., McKee, T.A., West, D.B.: The leafage of a chordal graph. Discuss. Math. Graph Theory 18(1), 23–48 (1998). https://doi.org/10.7151/dmgt.1061

    Article  MathSciNet  MATH  Google Scholar 

  18. Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth. SIAM J. Comput. 46(1), 161–189 (2017). https://doi.org/10.1137/140999980

    Article  MathSciNet  MATH  Google Scholar 

  19. 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

    Article  MathSciNet  MATH  Google Scholar 

  20. Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25(1), 42–65 (1982). https://doi.org/10.1016/0022-0000(82)90009-5

    Article  MathSciNet  MATH  Google Scholar 

  21. Neuen, D.: Hypergraph isomorphism for groups with restricted composition factors. In: Czumaj, A., Dawar, A., Merelli, E. (eds.) 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, Saarbrücken, Germany, 8–11 July 2020 (Virtual Conference). LIPIcs, vol. 168, pp. 88:1–88:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.ICALP.2020.88

  22. Ponomarenko, I.N.: The isomorphism problem for classes of graphs closed under contraction. J. Sov. Math. 55(2), 1621–1643 (1991)

    Article  Google Scholar 

  23. Ponomarenko, I.: Polynomial isomorphism algorithm for graphs which do not pinch to K\(_{3, g}\). J. Sov. Math. 34(4), 1819–1831 (1986)

    Article  Google Scholar 

  24. Schweitzer, P., Wiebking, D.: A unifying method for the design of algorithms canonizing combinatorial objects. In: Charikar, M., Cohen, E. (eds.) Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, 23–26 June 2019, pp. 1247–1258. ACM (2019)

    Google Scholar 

  25. Stacho, J.: On 2-subcolourings of chordal graphs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 544–554. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78773-0_47

    Chapter  Google Scholar 

  26. Weisfeiler, B., Leman, A.: The reduction of a graph to canonical form and the algebra which appears therein. NTI Series 2, 12–16 (1968). https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf. The URL links to an English translation

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Authors and Affiliations

  1. The Institute of Mathematical Sciences (HBNI), Chennai, India

    Vikraman Arvind

  2. Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Republic

    Roman Nedela

  3. V. A. Steklov Institue of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

    Ilia Ponomarenko

  4. Institut de mathématiques, Université de Neuchâtel, Neuchâtel, Switzerland

    Peter Zeman

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  1. Vikraman Arvind
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  2. Roman Nedela
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  3. Ilia Ponomarenko
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  4. Peter Zeman
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Correspondence to Peter Zeman .

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Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Arvind, V., Nedela, R., Ponomarenko, I., Zeman, P. (2022). Testing Isomorphism of Chordal Graphs of Bounded Leafage is Fixed-Parameter Tractable (Extended Abstract). In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_3

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