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Extended MSO Model Checking via Small Vertex Integrity

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Abstract

We study the model checking problem of an extended \(\textsf{MSO}\) with local and global cardinality constraints, called \(\textsf{MSO}^{\textsf{GL}}_{\textsf{Lin}}\), introduced recently by Knop et al. (Log Methods Comput Sci, 15(4), 2019. https://doi.org/10.23638/LMCS-15(4:12)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.

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Notes

  1. We assume that the readers are familiar with the concept of parameterized complexity. For standard definitions, see e.g., [16].

  2. In Ref. [20], Equitable Connected Partition was shown to be W[1]-hard parameterized simultaneously by pathwidth, feedback vertex set number, and the number of parts. In Sect. 6, we strengthen the W[1]-hardness by replacing pathwidth in the parameter with treedepth.

  3. The \(\textsf{MSO}_{2}\) logic is also known as the \(\textsf{GSO}\) logic, which stands for guarded second-order logic.

  4. To be more precise, we color the vertices in W with a new color to distinguish them from the original vertices.

  5. The parameter in [33] is a generalization of vertex integrity.

  6. In Sect. 6, we strengthen the hardness result by replacing this part with treedepth.

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Acknowledgements

The authors thank Michael Lampis and Valia Mitsou for fruitful discussions and sharing a preliminary version of Ref. [46].

Funding

This work was partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP18K11168, JP18K11169, JP20H05793, JP21K11752, JP22H00513.

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  1. Nagoya University, Nagoya, Japan

    Tatsuya Gima & Yota Otachi

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TG and YO wrote the main manuscript text.

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Preliminary version. A preliminary version appeared in the proceedings of the 33rd International Symposium on Algorithms and Computation (ISAAC 2022), Leibniz International Proceedings in Informatics 248 (2022) 20:1–21:15.

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Gima, T., Otachi, Y. Extended MSO Model Checking via Small Vertex Integrity. Algorithmica 86, 147–170 (2024). https://doi.org/10.1007/s00453-023-01161-9

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