Perfect Italian domination on planar and regular graphs

A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to {0,1,2}$ 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 $\gamma^pI(G)$, 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 $\gamma^pI(G) \leq c\mathcal{G} \times n$ 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 $\gamma^pI(G)$ 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 $k$.