# The complexity of the matching-cut problem for planar graphs and other graph classes

P.S. Bonsma

Research output: Book/ReportReportOther research output

### 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 Enschede University of Twente, Department of Applied Mathematics Published - 2002

### Publication series

Name Memorandum / Faculty of Mathematical Sciences Department of Applied Mathematics, University of Twente 1655 0169-2690

• 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.
Bonsma, P.S. / The complexity of the matching-cut problem for planar graphs and other graph classes. Enschede : University of Twente, Department of Applied Mathematics, 2002. (Memorandum / Faculty of Mathematical Sciences; 1655).
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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.",
keywords = "MSC-05C40, EWI-3475, IR-65841, MSC-68Q25",
author = "P.S. Bonsma",
note = "Imported from MEMORANDA",
year = "2002",
language = "Undefined",
series = "Memorandum / Faculty of Mathematical Sciences",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1655",

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Bonsma, PS 2002, The complexity of the matching-cut problem for planar graphs and other graph classes. Memorandum / Faculty of Mathematical Sciences, no. 1655, University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2002. (Memorandum / Faculty of Mathematical Sciences; No. 1655).

Research output: Book/ReportReportOther research output

TY - BOOK

T1 - The complexity of the matching-cut problem for planar graphs and other graph classes

AU - Bonsma, P.S.

N1 - Imported from MEMORANDA

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - MSC-05C40

KW - EWI-3475

KW - IR-65841

KW - MSC-68Q25

M3 - Report

T3 - Memorandum / Faculty of Mathematical Sciences

BT - The complexity of the matching-cut problem for planar graphs and other graph classes

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

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Bonsma PS. The complexity of the matching-cut problem for planar graphs and other graph classes. Enschede: University of Twente, Department of Applied Mathematics, 2002. (Memorandum / Faculty of Mathematical Sciences; 1655).