  The concept of Roman domination has recently been studied concerning
enumerating and counting (WG 2022). It has been shown that minimal Roman
dominating functions can be enumerated with polynomial delay, contrasting what
is known about minimal dominating sets. The running time of the algorithm could
be estimated as $\mathcal{O}(1.9332^n)$ on general graphs of order $n$. In this
paper, we focus on special graph classes. More specifically, for chordal
graphs, we present an enumeration algorithm running in time
$\mathcal{O}(1.8940^n)$. For interval graphs, we can lower this time further to
$\mathcal{O}(1.7321^n)$. Interestingly, this also matches (exactly) the known
lower bound. We can also provide a matching lower and upper bound for forests,
which is (incidentally) the same, namely $\mathcal{O}(\sqrt{3}^n)$.
Furthermore, we show an enumeration algorithm running in time
$\mathcal{O}(1.4656^n)$ for split graphs and for cobipartite graphs. Our
approach also allows to give concrete formulas for counting minimal Roman
dominating functions on special graph families like paths.
