  In a graph, a matching cut is an edge cut that is a matching. Matching Cut is
the problem of deciding whether or not a given graph has a matching cut, which
is known to be NP-complete even when restricted to bipartite graphs. It has
been proved that Matching Cut is polynomially solvable for graphs of diameter
two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut
is NP-complete in the class of graphs of diameter $d$. This resolves an open
problem posed by Borowiecki and Jesse-J\'ozefczyk in [Matching cutsets in
graphs of diameter $2$, Theoretical Computer Science 407 (2008) 574-582].
  We then show that, for any fixed integer $d\ge 4$, Matching Cut is
NP-complete even when restricted to the class of bipartite graphs of diameter
$d$. Complementing the hardness results, we show that Matching Cut is
polynomial-time solvable in the class of bipartite graphs of diameter at most
three, and point out a new and simple polynomial-time algorithm solving
Matching Cut in graphs of diameter $2$.
