### Abstract

The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Chv\'{a}tal studied this problem under the name of the Decomposable Graph Recognition problem, and proved the problem to be $\mathcal{NP}$-complete for graphs with maximum degree 4, and gave a polynomial algorithm for graphs with maximum degree 3. Recently, unaware of Chv\'{a}tal's result, Patrignani and Pizzonia also proved the $\mathcal{NP}$-completeness of the problem using a different reduction. They also posed the question whether the Matching-Cut problem is $\mathcal{NP}$-complete for planar graphs. In this paper an affirmative answer is given. Moreover, it is shown that the problem remains $\mathcal{NP}$-complete when restricted to planar bipartite graphs, planar graphs with girth 5 and planar graphs with maximum degree 4, making this the strongest result to date. The reduction is from Planar Graph 3-Colorability and differs from the reductions used to prove the earlier results.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Publication status | Published - 2002 |

### Publication series

Name | Memorandum / Faculty of Mathematical Sciences |
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Publisher | Department of Applied Mathematics, University of Twente |

No. | 1655 |

ISSN (Print) | 0169-2690 |

### Keywords

- MSC-05C40
- EWI-3475
- IR-65841
- MSC-68Q25

## Cite this

Bonsma, P. S. (2002).

*The complexity of the matching-cut problem for planar graphs and other graph classes*. (Memorandum / Faculty of Mathematical Sciences; No. 1655). Enschede: University of Twente, Department of Applied Mathematics.