  In a graph, a perfect matching cut is an edge cut that is a perfect matching.
Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has
a perfect matching cut, and is known to be NP-complete. We revisit the problem
and show that PMC remains NP-complete when restricted to bipartite graphs of
maximum degree 3 and arbitrarily large girth. Complementing this hardness
result, we give two graph classes in which PMC is polynomial time solvable. The
first one includes claw-free graphs and graphs without an induced path on five
vertices, the second one properly contains all chordal graphs. Assuming the
Exponential Time Hypothesis, we show there is no $O^*(2^{o(n)})$-time algorithm
for PMC even when restricted to $n$-vertex bipartite graphs, and also show that
PMC can be solved in $O^*(1.2721^n)$ time by means of an exact branching
algorithm.
