  The NP-complete problem Matching Cut is to decide if a graph has a matching
that is also an edge cut of the graph. We prove new complexity results for
Matching Cut restricted to $H$-free graphs, that is, graphs that do not contain
some fixed graph $H$ as an induced subgraph. We also prove new complexity
results for two recently studied variants of Matching Cut, on $H$-free graphs.
The first variant requires that the matching cut must be extendable to a
perfect matching of the graph. The second variant requires the matching cut to
be a perfect matching. In particular, we prove that there exists a small
constant $r>0$ such that the first variant is NP-complete for $P_r$-free
graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For
all three problems, we give state-of-the-art summaries of their computational
complexity for $H$-free graphs.
