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

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Bonsma, P.S. (2002) The complexity of the matching-cut problem for planar graphs and other graph classes. [Report]

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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.
Item Type:Report
Faculty:
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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Link to this item:http://purl.utwente.nl/publications/65841
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