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.
After revision:  they didn} 't provide an explicit bound on the running time , as it indirectly 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.