  One way to define the Matching Cut problem is: Given a graph $G$, is there an
edge-cut $M$ of $G$ such that $M$ is an independent set in the line graph of
$G$? We propose the more general Conflict-Free Cut problem: Together with the
graph $G$, we are given a so-called conflict graph $\hat{G}$ on the edges of
$G$, and we ask for an edge-cutset $M$ of $G$ that is independent in $\hat{G}$.
Since conflict-free settings are popular generalizations of classical
optimization problems and Conflict-Free Cut was not considered in the
literature so far, we start the study of the problem. We show that the problem
is $\textsf{NP}$-complete even when the maximum degree of $G$ is 5 and
$\hat{G}$ is 1-regular. The same reduction implies an exponential lower bound
on the solvability based on the Exponential Time Hypothesis. We also give
parameterized complexity results: We show that the problem is fixed-parameter
tractable with the vertex cover number of $G$ as a parameter, and we show
$\textsf{W[1]}$-hardness even when $G$ has a feedback vertex set of size one,
and the clique cover number of $\hat{G}$ is the parameter. Since the clique
cover number of $\hat{G}$ is an upper bound on the independence number of
$\hat{G}$ and thus the solution size, this implies $\textsf{W[1]}$-hardness
when parameterized by the cut size. We list polynomial-time solvable cases and
interesting open problems. At last, we draw a connection to a symmetric variant
of SAT.
