Extended MSO Model Checking via Small Vertex Integrity

Authors Tatsuya Gima , Yota Otachi

Thumbnail PDF


  • Filesize: 0.87 MB
  • 15 pages

Document Identifiers

Author Details

Tatsuya Gima
  • Nagoya University, Japan
Yota Otachi
  • Nagoya University, Japan


The authors thank Michael Lampis and Valia Mitsou for fruitful discussions and sharing a preliminary version of [Michael Lampis and Valia Mitsou, 2021].

Cite AsGet BibTex

Tatsuya Gima and Yota Otachi. Extended MSO Model Checking via Small Vertex Integrity. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We study the model checking problem of an extended MSO with local and global cardinality constraints, called MSO^GL_Lin, introduced recently by Knop, Koutecký, Masařík, and Toufar [Log. Methods Comput. Sci., 15(4), 2019]. We show that the problem is fixed-parameter tractable parameterized by vertex integrity, where vertex integrity is a graph parameter standing between vertex cover number and treedepth. Our result thus narrows the gap between the fixed-parameter tractability parameterized by vertex cover number and the W[1]-hardness parameterized by treedepth.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
  • vertex integrity
  • monadic second-order logic
  • cardinality constraint
  • fixed-parameter tractability


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. J. Algorithms, 12(2):308-340, 1991. URL: https://doi.org/10.1016/0196-6774(91)90006-K.
  2. Curtis A. Barefoot, Roger C. Entringer, and Henda C. Swart. Vulnerability in graphs - A comparative survey. J. Combin. Math. Combin. Comput., 1:13-22, 1987. Google Scholar
  3. Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi. Grundy distinguishes treewidth from pathwidth. In ESA 2020, volume 173 of LIPIcs, pages 14:1-14:19, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.14.
  4. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci., 209(1-2):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  5. Hans L. Bodlaender, Tesshu Hanaka, Yasuaki Kobayashi, Yusuke Kobayashi, Yoshio Okamoto, Yota Otachi, and Tom C. van der Zanden. Subgraph isomorphism on graph classes that exclude a substructure. Algorithmica, 82(12):3566-3587, 2020. URL: https://doi.org/10.1007/s00453-020-00737-z.
  6. Richard B. Borie, R. Gary Parker, and Craig A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7(5&6):555-581, 1992. URL: https://doi.org/10.1007/BF01758777.
  7. Laurent Bulteau, Konrad K. Dabrowski, Noleen Köhler, Sebastian Ordyniak, and Daniël Paulusma. An algorithmic framework for locally constrained homomorphisms. CoRR, abs/2201.11731, 2022. URL: http://arxiv.org/abs/2201.11731.
  8. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  9. Bruno Courcelle. The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. RAIRO Theor. Informatics Appl., 26:257-286, 1992. URL: https://doi.org/10.1051/ita/1992260302571.
  10. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach. Cambridge University Press, 2012. URL: https://www.cambridge.org/knowledge/isbn/item5758776/.
  11. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  12. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  13. Pål Grønås Drange, Markus S. Dregi, and Pim van 't Hof. On the computational complexity of vertex integrity and component order connectivity. Algorithmica, 76(4):1181-1202, 2016. URL: https://doi.org/10.1007/s00453-016-0127-x.
  14. Pavel Dvořák, Eduard Eiben, Robert Ganian, Dušan Knop, and Sebastian Ordyniak. Solving integer linear programs with a small number of global variables and constraints. In IJCAI 2017, pages 607-613, 2017. URL: https://doi.org/10.24963/ijcai.2017/85.
  15. Rosa Enciso, Michael R. Fellows, Jiong Guo, Iyad A. Kanj, Frances A. Rosamond, and Ondřej Suchý. What makes equitable connected partition easy. In IWPEC 2009, volume 5917 of Lecture Notes in Computer Science, pages 122-133, 2009. URL: https://doi.org/10.1007/978-3-642-11269-0_10.
  16. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Inf. Comput., 209(2):143-153, 2011. URL: https://doi.org/10.1016/j.ic.2010.11.026.
  17. Michael R. Fellows, Daniel Lokshtanov, Neeldhara Misra, Frances A. Rosamond, and Saket Saurabh. Graph layout problems parameterized by vertex cover. In ISAAC 2008, volume 5369 of Lecture Notes in Computer Science, pages 294-305, 2008. URL: https://doi.org/10.1007/978-3-540-92182-0_28.
  18. Jirí Fiala, Petr A. Golovach, and Jan Kratochvíl. Parameterized complexity of coloring problems: Treewidth versus vertex cover. Theor. Comput. Sci., 412(23):2513-2523, 2011. URL: https://doi.org/10.1016/j.tcs.2010.10.043.
  19. András Frank and Éva Tardos. An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica, 7:49-65, 1987. URL: https://doi.org/10.1007/BF02579200.
  20. Markus Frick and Martin Grohe. The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Log., 130(1-3):3-31, 2004. URL: https://doi.org/10.1016/j.apal.2004.01.007.
  21. Shinya Fujita and Michitaka Furuya. Safe number and integrity of graphs. Discret. Appl. Math., 247:398-406, 2018. URL: https://doi.org/10.1016/j.dam.2018.03.074.
  22. Shinya Fujita, Gary MacGillivray, and Tadashi Sakuma. Safe set problem on graphs. Discret. Appl. Math., 215:106-111, 2016. URL: https://doi.org/10.1016/j.dam.2016.07.020.
  23. Jakub Gajarský and Petr Hliněný. Kernelizing MSO properties of trees of fixed height, and some consequences. Log. Methods Comput. Sci., 11(1), 2015. URL: https://doi.org/10.2168/LMCS-11(1:19)2015.
  24. Robert Ganian. Twin-cover: Beyond vertex cover in parameterized algorithmics. In IPEC 2011, volume 7112 of Lecture Notes in Computer Science, pages 259-271, 2011. URL: https://doi.org/10.1007/978-3-642-28050-4_21.
  25. Robert Ganian, Petr Hlinený, Jaroslav Nesetril, Jan Obdrzálek, Patrice Ossona de Mendez, and Reshma Ramadurai. When trees grow low: Shrubs and fast MSO1. In MFCS 2012, volume 7464 of Lecture Notes in Computer Science, pages 419-430, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_38.
  26. Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On structural parameterizations of the bounded-degree vertex deletion problem. Algorithmica, 83(1):297-336, 2021. URL: https://doi.org/10.1007/s00453-020-00758-8.
  27. Robert Ganian and Jan Obdržálek. Expanding the expressive power of monadic second-order logic on restricted graph classes. In IWOCA 2013, volume 8288 of Lecture Notes in Computer Science, pages 164-177, 2013. URL: https://doi.org/10.1007/978-3-642-45278-9_15.
  28. Tatsuya Gima, Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, and Yota Otachi. Exploring the gap between treedepth and vertex cover through vertex integrity. In CIAC 2021, volume 12701 of Lecture Notes in Computer Science, pages 271-285, 2021. URL: https://doi.org/10.1007/978-3-030-75242-2_19.
  29. Martin Grohe and Stephan Kreutzer. Methods for algorithmic meta theorems. In Model Theoretic Methods in Finite Combinatorics, volume 558 of Contemporary Mathematics, pages 181-206, 2009. Google Scholar
  30. Petr Hliněný, Sang-il Oum, Detlef Seese, and Georg Gottlob. Width parameters beyond tree-width and their applications. Comput. J., 51(3):326-362, 2008. URL: https://doi.org/10.1093/comjnl/bxm052.
  31. Bart M. P. Jansen and Dániel Marx. Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and turing kernels. In SODA 2015, pages 616-629, 2015. URL: https://doi.org/10.1137/1.9781611973730.42.
  32. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12:415-440, 1987. URL: https://doi.org/10.1287/moor.12.3.415.
  33. Dušan Knop, Tomás Masarík, and Tomás Toufar. Parameterized complexity of fair vertex evaluation problems. In MFCS 2019, volume 138 of LIPIcs, pages 33:1-33:16, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.33.
  34. Dušan Knop, Martin Koutecký, Tomáš Masařík, and Tomáš Toufar. Simplified algorithmic metatheorems beyond MSO: Treewidth and neighborhood diversity. Log. Methods Comput. Sci., 15(4), 2019. URL: https://doi.org/10.23638/LMCS-15(4:12)2019.
  35. Stephan Kreutzer. Algorithmic meta-theorems. In Javier Esparza, Christian Michaux, and Charles Steinhorn, editors, Finite and Algorithmic Model Theory, volume 379 of London Mathematical Society Lecture Note Series, pages 177-270. Cambridge University Press, 2011. URL: https://doi.org/10.1017/cbo9780511974960.006.
  36. 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.
  37. Michael Lampis and Valia Mitsou. Fine-grained meta-theorems for vertex integrity. In ISAAC 2021, volume 212 of LIPIcs, pages 34:1-34:15, 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.34.
  38. Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL: https://doi.org/10.1287/moor.8.4.538.
  39. Li-Shin Lin and Sartaj Sahni. Fair edge deletion problems. IEEE Trans. Computers, 38(5):756-761, 1989. URL: https://doi.org/10.1109/12.24280.
  40. Tomáš Masařík and Tomáš Toufar. Parameterized complexity of fair deletion problems. Discret. Appl. Math., 278:51-61, 2020. URL: https://doi.org/10.1016/j.dam.2019.06.001.
  41. Stefan Szeider. Not so easy problems for tree decomposable graphs. Ramanujan Mathematical Society, Lecture Notes Series, No. 13:179-190, 2010. URL: http://arxiv.org/abs/1107.1177.
  42. Stefan Szeider. Monadic second order logic on graphs with local cardinality constraints. ACM Trans. Comput. Log., 12(2):12:1-12:21, 2011. URL: https://doi.org/10.1145/1877714.1877718.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail