In an undirected graph, a matching cut is an edge cut which is also a matching. we refer MATCHING CUT to the problem of deciding if a given graph contain a matching cut or not. For the matching cut problem, the size of the edge cut also known as the number of crossing edges is a natural parameter. Gomes et al. in \cite{Gomes-Sau} showed that MATCHING CUT is FPT when parameterized by maximum size of the edge cut using a reduction on results provided by Marx et. al \cite{marx_treewidth_reduction}. However, they didn't provide an explicit bound on the running time as the treewidth reduction technique of \cite{marx_treewidth_reduction} relies on a MSOL formulation for matching cut and Courcelle's Theorem \cite{courcelle1990monadic} to show fixed parameter tractability. This motivated us to design 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.
翻译:在未方向的图形中,匹配的剪切是一个匹配的边缘切切,它也是匹配的。我们将 Matching CUT 称为确定一个指定图表是否包含匹配的切切切。对于匹配的切问题,称为跨边缘数的边缘切断的大小也是一种自然参数。在\cite{Gomes-Sau}中的 Gomes 等人等人等人在\cite{Gomes-Sau} 中显示,当使用 Marx et 等等( ) al. al. 等 等公司提供的结果的减幅来根据最大边缘剪裁幅的参数大小参数进行参数的参数切切切切切切切切切切切切切切切切切切切割时,匹配。然而,他们并没有在运行的时间上提供明确的约束。然而,在本文中,我们提出了用于匹配切取参数依赖性明确的MACHTINKUT( MACHTIN CUT) 的FPT(FPT) 算法。