Before revision:  they didn}  't provide an explicit bound on the running time as the treewidth reduction technique of \mbox{%DIFAUXCMD marx_treewidth_reduction relies on a MSOL formulation for matching cut and Courcelle's Theorem \mbox{%DIFAUXCMD courcelle1990monadic an FPT algorithm for the MATCHING CUT where the dependence on the parameter is explicit. In this paper we present an FPT algorithm for matching cut problem for general graphs with running time O( 2^{O(k\log k)}n^{O(1)} ) . This is the first FPT algorithm for the MATCHING CUT where the dependence on the matching cut size as a parameter is explicit and bounded .
After revision:  they didn} 't provide an explicit bound on the running time as the treewidth reduction technique of \mbox{%DIFAUXCMD marx_treewidth_reduction relies on a MSOL formulation for matching cut and Courcelle's Theorem \mbox{%DIFAUXCMD courcelle1990monadic an FPT algorithm for the MATCHING CUT (and more generally for d-CUT) for general graphs with running time O( 2^{O(k\log k)}n^{O(1)} ) . This is the first FPT algorithm for the MATCHING CUT where the dependence on the matching cut size as a parameter is explicit and bounded .