Document Open Access Logo

Labelled Well Quasi Ordered Classes of Bounded Linear Clique-Width

Author Aliaume Lopez

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.MFCS.2025.70.pdf
  • Filesize: 1.09 MB
  • 17 pages
HTML5 Logo HTML (experimental)
LIPIcs.MFCS.2025.70.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Automata extensions
Keywords
  • well-quasi-ordering
  • linear clique-width
  • MSO transduction
  • automata theory

Metrics

Abstract

We construct an algorithm that inputs an MSO-interpretation from finite words to graphs, and decides if there exists a k ∈ ℕ such that the class of graphs induced by the interpretation is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using {1, …, k}. In case no such k exists, we also prove that the class of graphs is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using any well-quasi-ordered set of labels. As a byproduct of our analysis, we prove that for classes of bounded linear clique-width, a weak version of a conjecture by Pouzet holds.

Cite As Get BibTex

Aliaume Lopez. Labelled Well Quasi Ordered Classes of Bounded Linear Clique-Width. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.70

Author Details

Aliaume Lopez
  • University of Warsaw, Poland

Funding

  • Lopez, Aliaume: Aliaume Lopez was supported by the Polish National Science Centre (NCN) grant "Polynomial finite state computation" (2022/46/A/ST6/00072).

References

  1. Bogdan Alecu, Mamadou Moustapha Kanté, Vadim Lozin, and Viktor Zamaraev. Lettericity of graphs: an fpt algorithm and a bound on the size of obstructions, 2024. URL: https://doi.org/10.48550/arXiv.2402.12559.
  2. Aistis Atminas, Robert Brignall, Vadim V. Lozin, and Juraj Stacho. Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs. Discrete Applied Mathematics, 295:57-69, May 2021. URL: https://doi.org/10.1016/j.dam.2021.02.007.
  3. Robert Brignall, Michael Engen, and Vincent Vatter. A counterexample regarding labelled well-quasi-ordering. Graphs and Combinatorics, pages 1395-1409, 2018. URL: https://doi.org/10.1007/s00373-018-1962-0.
  4. Thomas Colcombet. A combinatorial theorem for trees. In Lars Arge, Christian Cachin, Tomasz Jurdziński, and Andrzej Tarlecki, editors, Automata, Languages and Programming, pages 901-912, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-73420-8_77.
  5. Thomas Colcombet. Green’s relations and their use in automata theory. In Adrian-Horia Dediu, Shunsuke Inenaga, and Carlos Martín-Vide, editors, Language and Automata Theory and Applications, pages 1-21, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-21254-3_1.
  6. Bruno Courcelle. Monadic second-order definable graph transductions: a survey. Theoretical Computer Science, 126(1):53-75, 1994. URL: https://doi.org/10.1016/0304-3975(94)90268-2.
  7. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. Journal of Computer and System Sciences, 46(2):218-270, 1993. URL: https://doi.org/10.1016/0022-0000(93)90004-G.
  8. Konrad K. Dabrowski, Vadim V. Lozin, and Daniël Paulusma. Well-quasi-ordering versus clique-width: New results on bigenic classes. Journal of Combinatorial Theory, Series B, 130:1-18, 2018. URL: https://doi.org/10.1016/j.jctb.2017.09.012.
  9. Jean Daligault, Michael Rao, and Stéphan Thomassé. Well-quasi-order of relabel functions. Order, 27(3):301-315, September 2010. URL: https://doi.org/10.1007/s11083-010-9174-0.
  10. Stéphane Demeri, Alain Finkel, Jean Goubault-Larrecq, Sylvain Schmitz, and Philippe Schnoebelen. Algorithmic aspects of wqo theory. Course notes for the MPRI Master’s program, Paris, 2012. URL: https://cel.archives-ouvertes.fr/cel-00727025.
  11. Nachum Derhowitz and Iddo Tzameret. Gap embedding for well-quasi-orderings1. Electronic Notes in Theoretical Computer Science, 84:80-90, 2003. WoLLIC'2003, 10th Workshop on Logic, Language, Information and Computation. URL: https://doi.org/10.1016/S1571-0661(04)80846-6.
  12. Guoli Ding. Subgraphs and well‐quasi‐ordering. Journal of Graph Theory, 16(5):489-502, November 1992. URL: https://doi.org/10.1002/jgt.3190160509.
  13. Samuel Eilenberg. Automata, Languages, and Machines, volume A of Automata, Languages, and Machines. Academic Press, 1974. URL: https://doi.org/10.5555/540244.
  14. Robert Ganian, Petr Hliněný, Jaroslav Nešetřil, Jan Obdržálek, and Patrice Ossona de Mendez. Shrub-depth: Capturing height of dense graphs. Logical Methods in Computer Science, Volume 15, Issue 1, January 2019. URL: https://doi.org/10.23638/LMCS-15(1:7)2019.
  15. Robert Ganian, Petr Hliněný, Jaroslav Nešetřil, Jan Obdržálek, Patrice Ossona de Mendez, and Reshma Ramadurai. When Trees Grow Low: Shrubs and Fast MSO1, pages 419-430. Springer Berlin Heidelberg, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_38.
  16. Christoph Haase, Sylvain Schmitz, and Philippe Schnoebelen. The power of priority channel systems. In Pedro R. D'Argenio and Hernán Melgratti, editors, CONCUR 2013 - Concurrency Theory, pages 319-333, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-40184-8_23.
  17. Graham Higman. Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society, 3:326-336, 1952. URL: https://doi.org/10.1112/plms/s3-2.1.326.
  18. Daniel Kirsten. The finite power problem revisited. Information Processing Letters, 84(6):291-294, 2002. URL: https://doi.org/10.1016/S0020-0190(02)00319-8.
  19. Joseph B. Kruskal. The theory of well-quasi-ordering: A frequently discovered concept. Journal of Combinatorial Theory, Series A, 13:297-305, 1972. URL: https://doi.org/10.1016/0097-3165(72)90063-5.
  20. Michal Kunc. Regular solutions of language inequalities and well quasi-orders. Theoretical Computer Science, 348(2):277-293, 2005. Automata, Languages and Programming: Algorithms and Complexity (ICALP-A 2004). URL: https://doi.org/10.1016/j.tcs.2005.09.018.
  21. Vadim V. Lozin, Igor Razgon, and Viktor Zamaraev. Well-quasi-ordering does not imply bounded clique-width. In Graph-Theoretic Concepts in Computer Science, pages 351-359, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-662-53174-7_25.
  22. Crispin St. John Alvah Nash-Williams. On well-quasi-ordering transfinite sequences. Mathematical Proceedings of the Cambridge Philosophical Society, 61:33-39, 1965. URL: https://doi.org/10.1017/S0305004100038603.
  23. Marko Petkovšek. Letter graphs and well-quasi-order by induced subgraphs. Discrete Mathematics, 244(1–3):375-388, February 2002. URL: https://doi.org/10.1016/s0012-365x(01)00094-2.
  24. Maurice Pouzet. Un bel ordre d'abritement et ses rapports avec les bornes d'une multirelation. CR Acad. Sci. Paris Sér. AB, 274:A1677-A1680, 1972. Google Scholar
  25. Neil Robertson and Paul D. Seymour. Graph minors. xx. wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. Special Issue Dedicated to Professor W.T. Tutte. URL: https://doi.org/10.1016/j.jctb.2004.08.001.
  26. Imre Simon. Factorization forests of finite height. Theoretical Computer Science, 72(1):65-94, 1990. URL: https://doi.org/10.1016/0304-3975(90)90047-L.
  27. Wolfgang Thomas. Languages, automata, and logic. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of formal languages, pages 389-455. Springer, 1997. URL: https://doi.org/10.1007/978-3-642-59136-5.
  28. Egon Wanke. k-nlc graphs and polynomial algorithms. Discrete Applied Mathematics, 54(2):251-266, 1994. URL: https://doi.org/10.1016/0166-218X(94)90026-4.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact