Matching cut: Kernelization, single-exponential time FPT, and exact exponential algorithms

Abstract In a graph, a matching cut is an edge cut that is a matching. Matching Cut , which is known to be NP-complete, is the problem of deciding whether or not a given graph  G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O ∗ ( 2 dc ( G ) ) -time and an O ∗ ( 2 d c ¯ ( G ) ) -time FPT algorithm for Matching Cut , where dc ( G ) and d c ¯ ( G ) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O ∗ ( 1 . 414 3 n ) to O ∗ ( 1 . 307 1 n ) where  n is the number of vertices in  G . Moreover, we point out that, unless NP ⊆ coNP ∕ poly , Matching Cut does not admit a polynomial kernel when parameterized simultaneously by treewidth, the number of edges crossing the cut, and the maximum degree of  G .

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