An FPT algorithm for Matching Cut and d-cut

Abstract

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 [7] showed that MATCH-ING CUT is FPT when parameterized by maximum size of the edge cut using a reduction on results provided by Marx et. al [14]. However, they didn't provide an explicit bound on the running time as the treewidth reduction technique of [14] relies on a MSOL formulation for matching cut and Courcelle's Theorem [5] 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.

The Perfect Matching Cut Problem Revisited

“…Further graph classes in which mc is polynomial time solvable were identified, such as graphs of bounded tree-width, claw-free, hole-free and Ore-graphs (see [4,7,20]). FPT algorithms and kernelization for mc with respect to various parameters has been discussed in [1,2,10,11,17,18]. The current-best exact algorithm solving mc has a running time of O * (1.3280 n ) where n is the vertex number of the input graph [17].…”