  The (Perfect) Matching Cut problem is to decide if a connected graph has a
(perfect) matching that is also an edge cut. The Disconnected Perfect Matching
problem is to decide if a connected graph has a perfect matching that contains
a matching cut. Both Matching Cut and Disconnected Perfect Matching are
NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known
to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed
girth. We prove that Matching Cut and Disconnected Perfect Matching are also
NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded
maximum degree. Our result for Matching Cut resolves a 20-year old open
problem. We also show that the more general problem $d$-Cut, for every fixed $d
\geq 1$, is NP-complete for bipartite graphs of arbitrarily large fixed girth
and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect
Matching Cut and Disconnected Perfect Matching are NP-complete for $H$-free
graphs whenever $H$ contains a connected component with two vertices of degree
at least 3. Afterwards, we update the state-of-the-art summaries for $H$-free
graphs and compare them with each other, and with a known and full
classification of the Maximum Matching Cut problem, which is to determine a
largest matching cut of a graph $G$. Finally, by combining existing results, we
obtain a complete complexity classification of Perfect Matching Cut for
$H$-subgraph-free graphs where $H$ is any finite set of graphs.
