  The (Perfect) Matching Cut problem is to decide if a graph $G$ has a
(perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of
$G$. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A
perfect matching cut is also a matching cut with maximum number of edges. To
increase our understanding of the relationship between the two problems, we
perform a complexity study for the Maximum Matching Cut problem, which is to
determine a largest matching cut in a graph. Our results yield full dichotomies
of Maximum Matching Cut for graphs of bounded diameter, bounded radius and
$H$-free graphs. A disconnected perfect matching of a graph $G$ is a perfect
matching that contains a matching cut of $G$. We also show how our new
techniques can be used for finding a disconnected perfect matching with a
largest matching cut for special graph classes. In this way we can prove that
the decision problem Disconnected Perfect Matching is polynomial-time solvable
for $(P_6+sP_2)$-free graphs for every $s\geq 0$, extending a known result for
$P_5$-free graphs (Bouquet and Picouleau, 2020).
