Before revision: 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.
After revision: In 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 , 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.