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Structural Parameterizations of Vertex Integrity [Best Paper]

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WALCOM: Algorithms and Computation (WALCOM 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14549))

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Abstract

The graph parameter vertex integrity measures how vulnerable a graph is to a removal of a small number of vertices. More precisely, a graph with small vertex integrity admits a small number of vertex removals to make the remaining connected components small. In this paper, we initiate a systematic study of structural parameterizations of the problem of computing the unweighted/weighted vertex integrity. As structural graph parameters, we consider well-known parameters such as clique-width, treewidth, pathwidth, treedepth, modular-width, neighborhood diversity, twin cover number, and cluster vertex deletion number. We show several positive and negative results and present sharp complexity contrasts.

Partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP20H00595, JP20H05793, JP20H05967, JP21H05852, JP21K11752, JP21K17707, JP21K19765, JP22H00513, JP23H03344, JP23KJ1066. The full version of this paper is available at http://arxiv.org/abs/2311.05892.

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Notes

  1. 1.

    We consider positive weights only since a vertex of non-positive weight is safely removed from the graph.

  2. 2.

    Note that this is the only part that requires the unary representation of weights. Note also that we cannot binary-search \(\ell \) as the irredundancy makes the problem non-monotone.

References

  1. Águeda, R., et al.: Safe sets in graphs: graph classes and structural parameters. J. Comb. Optim. 36(4), 1221–1242 (2018). https://doi.org/10.1007/s10878-017-0205-2

    Article  MathSciNet  Google Scholar 

  2. Bagga, K.S., Beineke, L.W., Goddard, W., Lipman, M.J., Pippert, R.E.: A survey of integrity. Discret. Appl. Math. 37(38), 13–28 (1992). https://doi.org/10.1016/0166-218X(92)90122-Q

    Article  MathSciNet  Google Scholar 

  3. Barefoot, C.A., Entringer, R.C., Swart, H.C.: Vulnerability in graphs – a comparative survey. J. Combin. Math. Combin. Comput. 1, 13–22 (1987)

    MathSciNet  Google Scholar 

  4. Belmonte, R., Hanaka, T., Katsikarelis, I., Lampis, M., Ono, H., Otachi, Y.: Parameterized complexity of safe set. J. Graph Algorithms Appl. 24(3), 215–245 (2020). https://doi.org/10.7155/jgaa.00528

    Article  MathSciNet  Google Scholar 

  5. Bentert, M., Heeger, K., Koana, T.: Fully polynomial-time algorithms parameterized by vertex integrity using fast matrix multiplication. In: ESA 2023. LIPIcs, vol. 274, pp. 16:1–16:16 (2023). https://doi.org/10.4230/LIPIcs.ESA.2023.16

  6. Clark, L.H., Entringer, R.C., Fellows, M.R.: Computational complexity of integrity. J. Combin. Math. Combin. Comput. 2, 179–191 (1987)

    MathSciNet  Google Scholar 

  7. Cygan, M., et al.: Parameterized Algorithms. Springer (2015). https://doi.org/10.1007/978-3-319-21275-3

  8. Doucha, M., Kratochvíl, J.: Cluster vertex deletion: a parameterization between vertex cover and clique-width. In: MFCS 2012. Lecture Notes in Computer Science, vol. 7464, pp. 348–359. Springer (2012). https://doi.org/10.1007/978-3-642-32589-2_32

  9. Drange, P.G., Dregi, M.S., van ’t Hof, P.: On the computational complexity of vertex integrity and component order connectivity. Algorithmica 76(4), 1181–1202 (2016). https://doi.org/10.1007/s00453-016-0127-x

  10. Dvořák, P., Eiben, E., Ganian, R., Knop, D., Ordyniak, S.: The complexity landscape of decompositional parameters for ILP: programs with few global variables and constraints. Artif. Intell. 300, 103561 (2021). https://doi.org/10.1016/j.artint.2021.103561

    Article  MathSciNet  Google Scholar 

  11. van Ee, M.: Some notes on bounded starwidth graphs. Inf. Process. Lett. 125, 9–14 (2017). https://doi.org/10.1016/j.ipl.2017.04.011

    Article  MathSciNet  Google Scholar 

  12. Fellows, M.R., Stueckle, S.: The immersion order, forbidden subgraphs and the complexity of network integrity. J. Combin. Math. Combin. Comput. 6, 23–32 (1989)

    MathSciNet  Google Scholar 

  13. Fujita, S., Furuya, M.: Safe number and integrity of graphs. Discret. Appl. Math. 247, 398–406 (2018). https://doi.org/10.1016/j.dam.2018.03.074

    Article  MathSciNet  Google Scholar 

  14. Gajarský, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: IPEC 2013. Lecture Notes in Computer Science, vol. 8246, pp. 163–176 (2013). https://doi.org/10.1007/978-3-319-03898-8_15

  15. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, W. H (1979)

    Google Scholar 

  16. Gima, T., Hanaka, T., Kiyomi, M., Kobayashi, Y., Otachi, Y.: Exploring the gap between treedepth and vertex cover through vertex integrity. Theor. Comput. Sci. 918, 60–76 (2022). https://doi.org/10.1016/j.tcs.2022.03.021

    Article  MathSciNet  Google Scholar 

  17. Gima, T., Otachi, Y.: Extended MSO model checking via small vertex integrity. In: ISAAC 2022. LIPIcs, vol. 248, pp. 20:1–20:15 (2022). https://doi.org/10.4230/LIPIcs.ISAAC.2022.20

  18. Hlinený, P., Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. Comput. J. 51(3), 326–362 (2008). https://doi.org/10.1093/comjnl/bxm052

    Article  Google Scholar 

  19. Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory Comput. Syst. 47(1), 196–217 (2010). https://doi.org/10.1007/s00224-008-9150-x

    Article  MathSciNet  Google Scholar 

  20. Kratsch, D., Kloks, T., Müller, H.: Measuring the vulnerability for classes of intersection graphs. Discret. Appl. Math. 77(3), 259–270 (1997). https://doi.org/10.1016/S0166-218X(96)00133-3

    Article  MathSciNet  Google Scholar 

  21. Lampis, M., Mitsou, V.: Fine-grained meta-theorems for vertex integrity. In: ISAAC 2021. LIPIcs, vol. 212, pp. 34:1–34:15 (2021). https://doi.org/10.4230/LIPIcs.ISAAC.2021.34

  22. Lee, E.: Partitioning a graph into small pieces with applications to path transversal. Math. Program. 177(1–2), 1–19 (2019). https://doi.org/10.1007/s10107-018-1255-7

    Article  MathSciNet  Google Scholar 

  23. Li, Y., Zhang, S., Zhang, Q.: Vulnerability parameters of split graphs. Int. J. Comput. Math. 85(1), 19–23 (2008). https://doi.org/10.1080/00207160701365721

    Article  MathSciNet  Google Scholar 

  24. McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discret. Math. 201(1–3), 189–241 (1999). https://doi.org/10.1016/S0012-365X(98)00319-7

    Article  MathSciNet  Google Scholar 

  25. Sorge, M., Weller, M.: The graph parameter hierarchy (2019). https://manyu.pro/assets/parameter-hierarchy.pdf

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Authors and Affiliations

  1. Nagoya University, Nagoya, Japan

    Tatsuya Gima, Ryota Murai, Hirotaka Ono & Yota Otachi

  2. JSPS Research Fellow, Tokyo, Japan

    Tatsuya Gima

  3. Kyushu University, Fukuoka, Japan

    Tesshu Hanaka

  4. Hokkaido University, Sapporo, Japan

    Yasuaki Kobayashi

Authors
  1. Tatsuya Gima
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  2. Tesshu Hanaka
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  3. Yasuaki Kobayashi
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  4. Ryota Murai
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  5. Hirotaka Ono
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  6. Yota Otachi
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Corresponding author

Correspondence to Yota Otachi .

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Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Japan

    Ryuhei Uehara

  2. Iwate University, Morioka, Japan

    Katsuhisa Yamanaka

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

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© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

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Gima, T., Hanaka, T., Kobayashi, Y., Murai, R., Ono, H., Otachi, Y. (2024). Structural Parameterizations of Vertex Integrity [Best Paper]. In: Uehara, R., Yamanaka, K., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2024. Lecture Notes in Computer Science, vol 14549. Springer, Singapore. https://doi.org/10.1007/978-981-97-0566-5_29

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  • DOI: https://doi.org/10.1007/978-981-97-0566-5_29

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