  The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing
each edge of $G$ exactly once and joining all these newly introduced vertices
of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is
a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that
every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which
$f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of
weights of vertices. The perfect Roman domination number is the minimum weight
of a perfect Roman dominating function on $G$. In this paper, we give a
characterization of middle graphs with equal Roman domination and perfect Roman
domination numbers.
