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Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius

Authors Felicia Lucke , Daniël Paulusma , Bernard Ries

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • matching cut
  • perfect matching
  • H-free graph
  • diameter
  • radius
  • dichotomy

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Abstract

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete, leading to many complexity results for both problems on special graph classes. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we introduce the Maximum Matching Cut problem. This problem is to determine a largest matching cut in a graph. We generalize and unify known polynomial-time algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of diameter at most 2 and to (P₆+sP₂)-free graphs. We also show that the complexity of Maximum Matching Cut differs from the complexities of Matching Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for 2P₃-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs.

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Felicia Lucke, Daniël Paulusma, and Bernard Ries. Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 64:1-64:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.MFCS.2023.64

Author Details

Felicia Lucke
  • Department of Informatics, University of Fribourg, Switzerland
Daniël Paulusma
  • Department of Computer Science, Durham University, UK
Bernard Ries
  • Department of Informatics, University of Fribourg, Switzerland

References

  1. Júlio Araújo, Nathann Cohen, Frédéric Giroire, and Frédéric Havet. Good edge-labelling of graphs. Discrete Applied Mathematics, 160:2502-2513, 2012. Google Scholar
  2. N. R. Aravind, Subrahmanyam Kalyanasundaram, and Anjeneya Swami Kare. Vertex partitioning problems on graphs with bounded tree width. Discrete Applied Mathematics, 319:254-270, 2022. Google Scholar
  3. N. R. Aravind and Roopam Saxena. An FPT algorithm for Matching Cut and d-Cut. Proc. IWOCA 2021, LNCS, 12757:531-543, 2021. Google Scholar
  4. Edouard Bonnet, Dibyayan Chakraborty, and Julien Duron. Cutting barnette graphs perfectly is hard. Proc. WG 2023, LNCS, to appear. Google Scholar
  5. Paul S. Bonsma. The complexity of the Matching-Cut problem for planar graphs and other graph classes. Journal of Graph Theory, 62:109-126, 2009 (conference version: WG 2003). Google Scholar
  6. Mieczyslaw Borowiecki and Katarzyna Jesse-Józefczyk. Matching cutsets in graphs of diameter 2. Theoretical Computer Science, 407:574-582, 2008. Google Scholar
  7. Valentin Bouquet and Christophe Picouleau. The complexity of the Perfect Matching-Cut problem. CoRR, abs/2011.03318, 2020. Google Scholar
  8. Chi-Yeh Chen, Sun-Yuan Hsieh, Hoàng-Oanh Le, Van Bang Le, and Sheng-Lung Peng. Matching Cut in graphs with large minimum degree. Algorithmica, 83:1238-1255, 2021. Google Scholar
  9. Maria Chudnovsky. The structure of bull-free graphs II and III - A summary. Journal of Combinatorial Theory, Series B, 102:252-282, 2012. Google Scholar
  10. Maria Chudnovsky and Paul Seymour. The structure of claw-free graphs. Surveys in Combinatorics, London Mathematical Society Lecture Note Series, 327:153-171, 2005. Google Scholar
  11. Vasek Chvátal. Recognizing decomposable graphs. Journal of Graph Theory, 8:51-53, 1984. Google Scholar
  12. Konrad K. Dabrowski, Matthew Johnson, and Daniël Paulusma. Clique-width for hereditary graph classes. Proc. BCC 2019, London Mathematical Society Lecture Note Series, 456:1-56, 2019. Google Scholar
  13. Arthur M. Farley and Andrzej Proskurowski. Networks immune to isolated line failures. Networks, 12:393-403, 1982. Google Scholar
  14. Carl Feghali. A note on Matching-Cut in P_t-free graphs. Information Processing Letters, 179:106294, 2023. Google Scholar
  15. Carl Feghali, Felicia Lucke, Daniël Paulusma, and Bernard Ries. Matching cuts in graphs of high girth and H-free graphs. CoRR, abs/2212.12317, 2022. Google Scholar
  16. Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of colouring graphs with forbidden subgraphs. Journal of Graph Theory, 84:331-363, 2017. Google Scholar
  17. Petr A. Golovach, Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Refined notions of parameterized enumeration kernels with applications to matching cut enumeration. Journal of Computer and System Sciences, 123:76-102, 2022. Google Scholar
  18. Petr A. Golovach, Daniël Paulusma, and Jian Song. Computing vertex-surjective homomorphisms to partially reflexive trees. Theoretical Computer Science, 457:86-100, 2012. Google Scholar
  19. Guilherme Gomes and Ignasi Sau. Finding cuts of bounded degree: complexity, FPT and exact algorithms, and kernelization. Algorithmica, 83:1677-1706, 2021. Google Scholar
  20. Ronald L. Graham. On primitive graphs and optimal vertex assignments. Annals of the New York Academy of Sciences, 175:170-186, 1970. Google Scholar
  21. Pinar Heggernes and Jan Arne Telle. Partitioning graphs into generalized dominating sets. Nordic Journal of Computing, 5:128-142, 1998. Google Scholar
  22. Danny Hermelin, Matthias Mnich, Erik Jan van Leeuwen, and Gerhard J. Woeginger. Domination when the stars are out. ACM Transactions on Algorithms, 15:25:1-25:90, 2019. Google Scholar
  23. Richard M. Karp. Reducibility among Combinatorial Problems. Complexity of Computer Computations, pages 85-103, 1972. Google Scholar
  24. Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Matching Cut: Kernelization, single-exponential time FPT, and exact exponential algorithms. Discrete Applied Mathematics, 283:44-58, 2020. Google Scholar
  25. Dieter Kratsch and Van Bang Le. Algorithms solving the Matching Cut problem. Theoretical Computer Science, 609:328-335, 2016. Google Scholar
  26. Hoang-Oanh Le and Van Bang Le. A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter. Theoretical Computer Science, 770:69-78, 2019. Google Scholar
  27. Hoàng-Oanh Le and Van Bang Le. Complexity results for matching cut problems in graphs without long induced paths. Proc. WG 2023, LNCS, to appear. Google Scholar
  28. Van Bang Le and Bert Randerath. On stable cutsets in line graphs. Theoretical Computer Science, 301:463-475, 2003. Google Scholar
  29. Van Bang Le and Jan Arne Telle. The Perfect Matching Cut problem revisited. Theoretical Computer Science, 931:117-130, 2022. Google Scholar
  30. Felicia Lucke, Daniël Paulusma, and Bernard Ries. On the complexity of Matching Cut for graphs of bounded radius and H-free graphs. Theoretical Computer Science, 936, 2022. Google Scholar
  31. Felicia Lucke, Daniël Paulusma, and Bernard Ries. Finding matching cuts in H-free graphs. Algorithmica, to appear. Google Scholar
  32. Augustine M. Moshi. Matching cutsets in graphs. Journal of Graph Theory, 13:527-536, 1989. Google Scholar
  33. Maurizio Patrignani and Maurizio Pizzonia. The complexity of the Matching-Cut problem. Proc. WG 2001, LNCS, 2204:284-295, 2001. Google Scholar
  34. Ján Plesník. Constrained weighted matchings and edge coverings in graphs. Discrete Applied Mathematics, 92:229-241, 1999. Google Scholar
  35. Bert Randerath and Ingo Schiermeyer. Vertex colouring and forbidden subgraphs - A survey. Graphs and Combinatorics, 20:1-40, 2004. Google Scholar
  36. Pim van’t Hof and Daniël Paulusma. A new characterization of P₆-free graphs. Discrete Applied Mathematics, 158:731-740, 2010. Google Scholar
  37. Mihalis Yannakakis. Node-and edge-deletion NP-complete problems. Proc. STOC 1978, pages 253-264, 1978. Google Scholar
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