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

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

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

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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.

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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.

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