  A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f :
V \to \{0,1,2\}$ such that for every vertex $f(v) = 0$, it holds that $\sum_{u
\in N(v)} f(u) = 2$, i.e., the weight of the labels assigned by $f$ to the
neighbors of $v$ is exactly two. The weight of a perfect Italian function is
the sum of the weights of the vertices. The perfect Italian domination number
of $G$, denoted by $\gamma^p_I(G)$, is the minimum weight of any perfect
Italian dominating function of $G$. While introducing the parameter, Haynes and
Henning (Discrete Appl. Math. (2019), 164--177) also proposed the problem of
determining the best possible constants $c_\mathcal{G}$ such that
$\gamma^p_I(G) \leq c_\mathcal{G} \times n$ for all graphs of order $n$ when
$G$ is in a particular class $\mathcal{G}$ of graphs. They proved that
$c_\mathcal{G} = 1$ when $\mathcal{G}$ is the class of bipartite graphs, and
raised the question for planar graphs and regular graphs. We settle their
question precisely for planar graphs by proving that $c_\mathcal{G} = 1$ and
for cubic graphs by proving that $c_\mathcal{G} = 2/3$. For split graphs, we
also show that $c_\mathcal{G} = 1$. In addition, we characterize the graphs $G$
with $\gamma^p_I(G)$ equal to 2 and 3 and determine the exact value of the
parameter for several simple structured graphs. We conclude by proving that it
is NP-complete to decide whether a given bipartite planar graph admits a
perfect Italian dominating function of weight $k$.
