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# Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization

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LIPIcs.IPEC.2019.19.pdf
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## Cite As

Guilherme C. M. Gomes and Ignasi Sau. Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2019.19

## Abstract

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call d-Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d-Cut on (2d+2)-regular graphs and a polynomial algorithm for graphs of maximum degree at most d+2. The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [IPEC 2018], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the distance to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O^*(2^n).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
• Mathematics of computing → Matchings and factors
##### Keywords
• matching cut
• bounded degree cut
• parameterized complexity
• FPT algorithm
• polynomial kernel
• distance to cluster

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## References

1. N. R. Aravind, Subrahmanyam Kalyanasundaram, and Anjeneya Swami Kare. On Structural Parameterizations of the Matching Cut Problem. In Proc. of the 11th International Conference on Combinatorial Optimization and Applications (COCOA), volume 10628 of LNCS, pages 475-482, 2017.
2. Amir Ban and Nati Linial. Internal partitions of regular graphs. Journal of Graph Theory, 83(1):5-18, 2016.
3. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009.
4. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Cross-Composition: A New Technique for Kernelization Lower Bounds. In Proc. of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 9 of LIPIcs, pages 165-176, 2011.
5. Paul S. Bonsma. The complexity of the matching-cut problem for planar graphs and other graph classes. Journal of Graph Theory, 62(2):109-126, 2009.
6. Anudhyan Boral, Marek Cygan, Tomasz Kociumaka, and Marcin Pilipczuk. A fast branching algorithm for cluster vertex deletion. Theory of Computing Systems, 58(2):357-376, 2016.
7. Vasek Chvátal. Recognizing decomposable graphs. Journal of Graph Theory, 8(1):51-53, 1984.
8. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and computation, 85(1):12-75, 1990.
9. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
10. Matt DeVos. http://www.openproblemgarden.org/op/friendly_partitions, 2009.
11. Reinhard Diestel. Graph Theory, volume 173. Springer-Verlag, 4th edition, 2010.
12. R. G. Downey and M. R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
13. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2019.
14. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. Journal of Computer and System Sciences, 77(1):91-106, 2011.
15. Ron L Graham. On primitive graphs and optimal vertex assignments. Annals of the New York academy of sciences, 175(1):170-186, 1970.
16. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001.
17. Atsushi Kaneko. On decomposition of triangle-free graphs under degree constraints. Journal of Graph Theory, 27(1):7-9, 1998.
18. Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Matching Cut: Kernelization, Single-Exponential Time FPT, and Exact Exponential Algorithms. In Proc. of the 13th International Symposium on Parameterized and Exact Computation (IPEC), volume 115 of LIPIcs, pages 19:1-19:13, 2018.
19. Dieter Kratsch and Van Bang Le. Algorithms solving the matching cut problem. Theoretical Computer Science, 609:328-335, 2016.
20. Hoàng-Oanh Le and Van Bang Le. On the Complexity of Matching Cut in Graphs of Fixed Diameter. In Proc. of the 27th International Symposium on Algorithms and Computation (ISAAC), volume 64 of LIPIcs, pages 50:1-50:12, 2016.
21. Van Bang Le and Bert Randerath. On stable cutsets in line graphs. Theoretical Computer Science, 301(1-3):463-475, 2003.
22. Jie Ma and Tianchi Yang. Decomposing C₄-free graphs under degree constraints. Journal of Graph Theory, 90(1):13-23, 2019.
23. Dániel Marx, Barry O'Sullivan, and Igor Razgon. Treewidth Reduction for Constrained Separation and Bipartization Problems. In Proc. of the 27th International Symposium on Theoretical Aspects of Computer Science, (STACS), volume 5 of LIPIcs, pages 561-572, 2010.
24. Augustine M Moshi. Matching cutsets in graphs. Journal of Graph Theory, 13(5):527-536, 1989.
25. Maurizio Patrignani and Maurizio Pizzonia. The complexity of the matching-cut problem. In Proc. of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 2204 of LNCS, pages 284-295, 2001.
26. Khurram H. Shafique and Ronald D. Dutton. On satisfactory partitioning of graphs. Congressus Numerantium, pages 183-194, 2002.
27. Michael Stiebitz. Decomposing graphs under degree constraints. Journal of Graph Theory, 23(3):321-324, 1996.
28. Carsten Thomassen. Graph decomposition with constraints on the connectivity and minimum degree. Journal of Graph Theory, 7(2):165-167, 1983.
29. Chee-Keng Yap. Some Consequences of Non-Uniform Conditions on Uniform Classes. Theoretical Computer Science, 26:287-300, 1983.
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