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# Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity

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## Cite As

Sriram Bhyravarapu, Satyabrata Jana, Saket Saurabh, and Roohani Sharma. Difference Determines the Degree: Structural Kernelizations of Component Order Connectivity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.5

## Abstract

We consider the question of polynomial kernelization of a generalization of the classical Vertex Cover problem parameterized by a parameter that is provably smaller than the solution size. In particular, we focus on the c-Component Order Connectivity problem (c-COC) where given an undirected graph G and a non-negative integer t, the objective is to test whether there exists a set S of size at most t such that every component of G-S contains at most c vertices. Such a set S is called a c-coc set. It is known that c-COC admits a kernel with {O}(ct) vertices. Observe that for c = 1, this corresponds to the Vertex Cover problem. We study the c-Component Order Connectivity problem parameterized by the size of a d-coc set (c-COC/d-COC), where c,d ∈ ℕ with c ≤ d. In particular, the input is an undirected graph G, a positive integer t and a set M of at most k vertices of G, such that the size of each connected component in G - M is at most d. The question is to find a set S of vertices of size at most t, such that the size of each connected component in G - S is at most c. In this paper, we give a kernel for c-COC/d-COC with O(k^{d-c+1}) vertices and O(k^{d-c+2}) edges. Our result exhibits that the difference in d and c, and not their absolute values, determines the exact degree of the polynomial in the kernel size. When c = d = 1, the c-COC/d-COC problem is exactly the Vertex Cover problem parameterized by the solution size, which has a kernel with O(k) vertices and O(k²) edges, and this is asymptotically tight [Dell & Melkebeek, JACM 2014]. We also show that the dependence of d-c in the exponent of the kernel size cannot be avoided under reasonable complexity assumptions.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Fixed parameter tractability
##### Keywords
• Kernelization
• Component Order Connectivity
• Vertex Cover
• Structural Parameterizations

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## References

1. Marin Bougeret, Bart M. P. Jansen, and Ignasi Sau. Bridge-depth characterizes which minor-closed structural parameterizations of vertex cover admit a polynomial kernel. SIAM J. Discret. Math., 36(4):2737-2773, 2022. URL: https://doi.org/10.1137/21m1400766.
2. Marin Bougeret and Ignasi Sau. How much does a treedepth modulator help to obtain polynomial kernels beyond sparse graphs? Algorithmica, 81(10):4043-4068, 2019. URL: https://doi.org/10.1007/s00453-018-0468-8.
3. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
4. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
5. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM J. Discret. Math., 30(1):383-410, 2016. URL: https://doi.org/10.1137/140997889.
6. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing: Theory of parameterized preprocessing. Cambridge University Press, United Kingdom, 2019. Publisher Copyright: copyright Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi 2019. URL: https://doi.org/10.1017/9781107415157.
7. Fedor V. Fomin and Torstein J. F. Strømme. Vertex cover structural parameterization revisited. In Pinar Heggernes, editor, Graph-Theoretic Concepts in Computer Science - 42nd International Workshop, WG 2016, Istanbul, Turkey, June 22-24, 2016, Revised Selected Papers, volume 9941 of Lecture Notes in Computer Science, pages 171-182, 2016. URL: https://doi.org/10.1007/978-3-662-53536-3_15.
8. Archontia C. Giannopoulou, Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. Uniform kernelization complexity of hitting forbidden minors. ACM Trans. Algorithms, 13(3):35:1-35:35, 2017. URL: https://doi.org/10.1145/3029051.
9. Eva-Maria C. Hols and Stefan Kratsch. Smaller parameters for vertex cover kernelization. In Daniel Lokshtanov and Naomi Nishimura, editors, 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, September 6-8, 2017, Vienna, Austria, volume 89 of LIPIcs, pages 20:1-20:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.20.
10. Bart M. P. Jansen and Hans L. Bodlaender. Vertex cover kernelization revisited - upper and lower bounds for a refined parameter. Theory Comput. Syst., 53(2):263-299, 2013. URL: https://doi.org/10.1007/s00224-012-9393-4.
11. Stefan Kratsch. A randomized polynomial kernelization for vertex cover with a smaller parameter. SIAM J. Discret. Math., 32(3):1806-1839, 2018. URL: https://doi.org/10.1137/16M1104585.
12. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. J. ACM, 67(3):16:1-16:50, 2020. URL: https://doi.org/10.1145/3390887.
13. Mithilesh Kumar and Daniel Lokshtanov. A 2lk kernel for l-component order connectivity. In Jiong Guo and Danny Hermelin, editors, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, volume 63 of LIPIcs, pages 20:1-20:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.20.
14. Michael Lampis. A kernel of order 2 k-c log k for vertex cover. Inf. Process. Lett., 111(23-24):1089-1091, 2011. URL: https://doi.org/10.1016/j.ipl.2011.09.003.
15. Diptapriyo Majumdar, Venkatesh Raman, and Saket Saurabh. Polynomial kernels for vertex cover parameterized by small degree modulators. Theory Comput. Syst., 62(8):1910-1951, 2018. URL: https://doi.org/10.1007/s00224-018-9858-1.
16. Geevarghese Philip, Varun Rajan, Saket Saurabh, and Prafullkumar Tale. Subset feedback vertex set in chordal and split graphs. Algorithmica, 81(9):3586-3629, 2019. URL: https://doi.org/10.1007/s00453-019-00590-9.
17. Stéphan Thomassé. A 4k^2 kernel for feedback vertex set. ACM Trans. Algorithms, 6(2):32:1-32:8, 2010. URL: https://doi.org/10.1145/1721837.1721848.
18. Mingyu Xiao. Linear kernels for separating a graph into components of bounded size. J. Comput. Syst. Sci., 88:260-270, 2017. URL: https://doi.org/10.1016/j.jcss.2017.04.004.