The Complexity of the Matching-Cut Problem for Planar Graphs and Other Graph Classes

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 NP-complete for graphs with maximum degree 4, and gave a polynomial algorithm for graphs with maximum degree 3. Recently, unaware of Chvátal’s result, Patrignani and Pizzonia also proved the NP-completeness of the problem using a different reduction. They also posed the question whether the Matching-Cut problem is NP-complete for planar graphs. In this paper an affirmative answer is given. Moreover, it is shown that the problem remains 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.