Document Open Access Logo

Matching Cuts in Graphs of High Girth and H-Free Graphs

Authors Carl Feghali , Felicia Lucke , Daniël Paulusma , Bernard Ries

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2023.31.pdf
  • Filesize: 0.76 MB
  • 16 pages

Document Identifiers

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed d ≥ 1, is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for H-subgraph-free graphs where H is any finite set of graphs.

Cite As Get BibTex

Carl Feghali, Felicia Lucke, Daniël Paulusma, and Bernard Ries. Matching Cuts in Graphs of High Girth and H-Free Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.31

Author Details

Carl Feghali
  • University of Lyon, EnsL, CNRS, LIP, F-69342, Lyon Cedex 07, France
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

Acknowledgements

We thank Hoang-Oanh Le for a significant simplification of our original proof of Theorem 6, which we simplified a bit further. We thank Van Bang Le for observing the bound on the maximum degree in Theorem 6, solving an Open Problem Garden question.

References

  1. Noga Alon and Vitali D Milman. λ1, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38:73-88, 1985. Google Scholar
  2. Júlio Araújo, Nathann Cohen, Frédéric Giroire, and Frédéric Havet. Good edge-labelling of graphs. Discrete Applied Mathematics, 160:2502-2513, 2012. Google Scholar
  3. 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
  4. 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
  5. Edouard Bonnet, Dibyayan Chakraborty, and Julien Duron. Cutting barnette graphs perfectly is hard. Proc. WG 2023, LNCS, to appear. Google Scholar
  6. 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
  7. Mieczyslaw Borowiecki and Katarzyna Jesse-Józefczyk. Matching cutsets in graphs of diameter 2. Theoretical Computer Science, 407:574-582, 2008. Google Scholar
  8. Valentin Bouquet and Christophe Picouleau. The complexity of the Perfect Matching-Cut problem. CoRR, abs/2011.03318, 2020. URL: https://arxiv.org/abs/2011.03318.
  9. 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
  10. Vasek Chvátal. Recognizing decomposable graphs. Journal of Graph Theory, 8:51-53, 1984. Google Scholar
  11. Peter G. Lejeune Dirichlet. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften zu Berlin, 1837. Google Scholar
  12. Jozef Dodziuk. Difference equations, isoperimetric inequality and transience of certain random walks. Transactions of the American Mathematical Society, 284:787-794, 1984. 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. 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
  16. 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
  17. Guilherme Gomes and Ignasi Sau. Finding cuts of bounded degree: complexity, FPT and exact algorithms, and kernelization. Algorithmica, 83:1677-1706, 2021. Google Scholar
  18. Ronald L. Graham. On primitive graphs and optimal vertex assignments. Annals of the New York Academy of Sciences, 175:170-186, 1970. Google Scholar
  19. Pinar Heggernes and Jan Arne Telle. Partitioning graphs into generalized dominating sets. Nordic Journal of Computing, 5:128-142, 1998. Google Scholar
  20. Matthew Johnson, Barnaby Martin, Jelle J. Oostveen, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs I: The framework. CoRR, abs/2211.12887, 2022. URL: https://arxiv.org/abs/2211.12887.
  21. Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity framework for forbidden subgraphs III: When problems are tractable on subcubic graphs. Proc. MFCS 2023, LIPIcs, 272:57:1-57:15, 2023. Google Scholar
  22. 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
  23. Dieter Kratsch and Van Bang Le. Algorithms solving the Matching Cut problem. Theoretical Computer Science, 609:328-335, 2016. Google Scholar
  24. 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
  25. 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
  26. Van Bang Le and Bert Randerath. On stable cutsets in line graphs. Theoretical Computer Science, 301:463-475, 2003. Google Scholar
  27. Van Bang Le and Jan Arne Telle. The Perfect Matching Cut problem revisited. Proc. WG 2021, LNCS, 12911:182-194, 2021. Google Scholar
  28. Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8:261-277, 1988. Google Scholar
  29. 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
  30. Felicia Lucke, Daniël Paulusma, and Bernard Ries. Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius. Proc. MFCS 2023, LIPIcs, 272:64:1-64:15, 2023. 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. Open problem garden. http://www.openproblemgarden.org/op/matching_cut_and_girth. (accessed on 22 June 2023).
  34. Maurizio Patrignani and Maurizio Pizzonia. The complexity of the Matching-Cut problem. Proc. WG 2001, LNCS, 2204:284-295, 2001. Google Scholar
  35. Harold N. Shapiro. Introduction to the Theory of Numbers. Dover Publications, 2008. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact