Document Open Access Logo

ℋ-Clique-Width and a Hereditary Analogue of Product Structure

Authors Petr Hliněný , Jan Jedelský

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2024.61.pdf
  • Filesize: 0.68 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • product structure
  • hereditary class
  • clique-width
  • twin-width

Metrics

Abstract

We introduce a novel generalization of the notion of clique-width which aims to bridge the gap between classical hereditary width measures and the recently introduced graph product structure theory. Bounding the new H-clique-width, in the special case of H being the class of paths, is equivalent to admitting a hereditary (i.e., induced) product structure of a path times a graph of bounded clique-width. Furthermore, every graph admitting the usual (non-induced) product structure of a path times a graph of bounded tree-width, has bounded H-clique-width and, as a consequence, it admits the usual product structure in an induced way. We prove further basic properties of H-clique-width in general.

Cite As Get BibTex

Petr Hliněný and Jan Jedelský. ℋ-Clique-Width and a Hereditary Analogue of Product Structure. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.61

Author Details

Petr Hliněný
  • Masaryk University, Brno, Czech Republic
Jan Jedelský
  • Masaryk University, Brno, Czech Republic

Acknowledgements

We acknowledge helpful discussions with Jakub Gajarský on the questions and problems posed in Section 5, and especially on the question of H-clique-width possibly being complement-closed.

References

  1. Marthe Bonamy, Cyril Gavoille, and Michal Pilipczuk. Shorter labeling schemes for planar graphs. SIAM J. Discret. Math., 36(3):2082-2099, 2022. URL: https://doi.org/10.1137/20M1330464.
  2. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. In SODA, pages 1977-1996. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.118.
  3. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. J. ACM, 69(1):3:1-3:46, 2022. URL: https://doi.org/10.1145/3486655.
  4. Édouard Bonnet, O-joung Kwon, and David R. Wood. Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond). CoRR, abs/2202.11858, 2022. URL: https://arxiv.org/abs/2202.11858.
  5. Anuj Dawar, Martin Grohe, and Stephan Kreutzer. Locally excluding a minor. In LICS, pages 270-279. IEEE Computer Society, 2007. URL: https://doi.org/10.1109/LICS.2007.31.
  6. Patrice Ossona de Mendez. First-order transductions of graphs (invited talk). In STACS, volume 187 of LIPIcs, pages 2:1-2:7. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.STACS.2021.2.
  7. Guoli Ding, Bogdan Oporowski, James G. Oxley, and Dirk Vertigan. Unavoidable minors of large 3-connected binary matroids. J. Comb. Theory, Ser. B, 66(2):334-360, 1996. URL: https://doi.org/10.1006/JCTB.1996.0026.
  8. Jan Dreier, Nikolas Mählmann, and Szymon Torunczyk. Flip-breakability: A combinatorial dichotomy for monadically dependent graph classes. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 1550-1560. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649739.
  9. Vida Dujmovic, Louis Esperet, Cyril Gavoille, Gwenaël Joret, Piotr Micek, and Pat Morin. Adjacency labelling for planar graphs (and beyond). J. ACM, 68(6):42:1-42:33, 2021. URL: https://doi.org/10.1145/3477542.
  10. Vida Dujmovic, Louis Esperet, Gwenaël Joret, Bartosz Walczak, and David R. Wood. Planar graphs have bounded nonrepetitive chromatic number. Advances in Combinatorics, March 2020. URL: https://doi.org/10.19086/aic.12100.
  11. Vida Dujmovic, Gwenaël Joret, Piotr Micek, Pat Morin, Torsten Ueckerdt, and David R. Wood. Planar graphs have bounded queue-number. J. ACM, 67(4):22:1-22:38, 2020. URL: https://doi.org/10.1145/3385731.
  12. Vida Dujmovic, Pat Morin, and David R. Wood. Graph product structure for non-minor-closed classes. J. Comb. Theory, Ser. B, 162:34-67, 2023. URL: https://doi.org/10.1016/J.JCTB.2023.03.004.
  13. Zdenek Dvořák, Tony Huynh, Gwenaël Joret, Chun-Hung Liu, and David R. Wood. Notes on graph product structure theory. In 2019-20 MATRIX Annals, pages 513-533, Cham, 2021. Springer International Publishing. Google Scholar
  14. Michael R. Fellows, Frances A. Rosamond, Udi Rotics, and Stefan Szeider. Clique-width is NP-complete. SIAM J. Discret. Math., 23(2):909-939, 2009. URL: https://doi.org/10.1137/070687256.
  15. Markus Frick and Martin Grohe. Deciding first-order properties of locally tree-decomposable structures. J. ACM, 48(6):1184-1206, 2001. URL: https://doi.org/10.1145/504794.504798.
  16. Petr Hliněný and Jan Jedelský. Twin-width of planar graphs is at most 8, and at most 6 when bipartite planar. In ICALP, volume 261 of LIPIcs, pages 75:1-75:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.75.
  17. Petr Hliněný and Jan Jedelský. H-clique-width and a hereditary analogue of product structure. CoRR, abs/2403.16789, 2024. URL: https://arxiv.org/abs/2403.16789.
  18. Petr Hliněný and Sang-il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput., 38(3):1012-1032, 2008. URL: https://doi.org/10.1137/070685920.
  19. Hugo Jacob and Marcin Pilipczuk. Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs. In WG, volume 13453 of Lecture Notes in Computer Science, pages 287-299. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-15914-5_21.
  20. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012. URL: https://doi.org/10.1007/S00453-011-9554-X.
  21. Sang-il Oum and Paul Seymour. Approximating clique-width and branch-width. J. Comb. Theory, Ser. B, 96(4):514-528, 2006. URL: https://doi.org/10.1016/J.JCTB.2005.10.006.
  22. D. Seese. Linear time computable problems and first-order descriptions. Math. Structures Comput. Sci., 6(6):505-526, 1996. URL: https://doi.org/10.1016/S1571-0661(05)80203-8.
  23. Torsten Ueckerdt, David R. Wood, and Wendy Yi. An improved planar graph product structure theorem. Electron. J. Comb., 29(2), 2022. URL: https://doi.org/10.37236/10614.
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