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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We consider positive weights only since a vertex of non-positive weight is safely removed from the graph.
- 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
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
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
Barefoot, C.A., Entringer, R.C., Swart, H.C.: Vulnerability in graphs – a comparative survey. J. Combin. Math. Combin. Comput. 1, 13–22 (1987)
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
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
Clark, L.H., Entringer, R.C., Fellows, M.R.: Computational complexity of integrity. J. Combin. Math. Combin. Comput. 2, 179–191 (1987)
Cygan, M., et al.: Parameterized Algorithms. Springer (2015). https://doi.org/10.1007/978-3-319-21275-3
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
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
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
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
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)
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
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
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, W. H (1979)
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
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
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
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
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
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
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
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
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
Sorge, M., Weller, M.: The graph parameter hierarchy (2019). https://manyu.pro/assets/parameter-hierarchy.pdf
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-981-97-0566-5_29
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-0565-8
Online ISBN: 978-981-97-0566-5
eBook Packages: Computer ScienceComputer Science (R0)