Perfect Italian domination on planar and regular graphs

A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to {0,1,2}$suchthatforeveryvertex$f(v) = 0$,itholdsthat$\sum_{u \in N(v)} f(u) = 2$,i\mathrm{.}e\mathrm{.},theweightofthelabelsassignedby$f$ 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 ${\gamma }_I^p(G)$, 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 ${\gamma }_I^p(G)\le c_{\mathcal{G}}\times n$ 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 $c_{\mathcal{G}}=2\mathrm{/}3$. For split graphs, we also show that $c_{\mathcal{G}}=1$. In addition, we characterize the graphs $G$ with ${\gamma }_I^p(G)$ 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 $k$.