Abstract
The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Chvátal 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. In this paper it is shown that the problem is \(\mathcal{NP}\)-complete when restricted to planar graphs with girth 5 and planar graphs with maximum degree 4. In addition, for claw-free graphs and planar graphs with girth at least 7 polynomial algorithms to find matching-cuts are described.
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Bonsma, P. (2003). The Complexity of the Matching-Cut Problem for Planar Graphs and Other Graph Classes. In: Bodlaender, H.L. (eds) Graph-Theoretic Concepts in Computer Science. WG 2003. Lecture Notes in Computer Science, vol 2880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39890-5_9
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DOI: https://doi.org/10.1007/978-3-540-39890-5_9
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