  Given a positive integer $d$, the d-CUT is the problem of deciding if an
undirected graph $G=(V,E)$ has a cut $(A,B)$ such that every vertex in $A$
(resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). For $d=1$, the
problem is referred to as MATCHING CUT. Gomes and Sau, in IPEC 2019, gave the
first fixed parameter tractable algorithm for d-CUT parameterized by maximum
number of the crossing edges in the cut (i.e. the size of edge cut). However,
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 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 d-CUT and MATCHING CUT with an
explicit dependence on this parameter. We also observe that there is no
algorithm solving MATCHING CUT in time $2^{o(k)}n^{O(1)}$ where $k$ is the
maximum size of the edge cut unless ETH fails.
