In an undirected graph  , a matching cut is an edge cut which is also a matching. we refer MATCHING CUT to the problemof 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 \mbox{%DIFAUXCMD Gomes-Sau0pt%DIFAUXCMD showed that MATCHING CUT is FPT when parameterized by maximum size of the edge cutusing a reduction on results provided by Marx et. al \mbox{%DIFAUXCMD marx_treewidth_reduction}0pt%DIFAUXCMD . However, 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 .
Given a positive integer d, the d-CUT problem is to decide if an undirected graph G=(V,E) has a non trivial bipartition (A,B) of V such that every vertex in A (resp. B) has at most d neighbors in B (resp. A). When d=1, this is the MATCHING CUT problem. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for d-CUT, when parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However , 0pt%DIFAUXCMD showed that MATCHING CUT is FPT when parameterized by maximum size of the edge cutusing a reduction on results provided by Marx et. al \mbox{%DIFAUXCMD marx_treewidth_reduction}0pt%DIFAUXCMD . However, they didn} their paper doesn 't provide an explicit bound on the running time , as it indirectly relies on a MSOL formulation and Courcelle's Theorem . Motivated by this, we design and present an FPT algorithm for the MATCHING CUT (and more generally for d-CUT) for general graphs with running time 2^{O(k\log k)}n^{O(1)} where k is the maximum size of the edge cut . This is the first FPT algorithm for the MATCHING CUT (and d-CUT) with an explicit dependence on this parameter. We also observe a lower bound of 2^{\Omega(k) .
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Given a positive integer d, the d-CUT problem is to decide if
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G=(V,E) has a non trivial bipartition (A,B) of V such that every vertex in A (resp. B) has at most d neighbors in B (resp. A). When d=1, this is the MATCHING CUT problem. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for d-CUT, when parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However
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a matching cut is an edge cut which is also a matching. we refer MATCHING CUT to the problemof 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 \mbox{%DIFAUXCMD Gomes-Sau
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their paper doesn
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as the treewidth reduction technique of \mbox{%DIFAUXCMD marx_treewidth_reduction
, as it indirectly
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for matching cut
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\mbox{%DIFAUXCMD courcelle1990monadic
. Motivated by this, we design and present
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where the dependence on the parameter is explicit. In this paper we present an FPT algorithm for matching cut problem
(and more generally for d-CUT)
D
O(
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)
where k is the maximum size of the edge cut
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where the dependence on the matching cut size as a parameter is explicit and bounded
(and d-CUT) with an explicit dependence on this parameter. We also observe a lower bound of 2^{\Omega(k)