From Italian domination in lexicographic product graphs to w-domination in graphs

In this paper, we show that the Italian domination number of every lexicographic product graph $G\circ H$ can be expressed in terms of five different domination parameters of $G$. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let $N(v)$ denote the open neighbourhood of $v\in V(G)$, and let $w=(w_0,w_1, \dots,w_l)$ be a vector of nonnegative integers such that $ w_0\ge 1$. We say that a function $f: V(G)\longrightarrow \{0,1,\dots ,l\}$ is a $w$-dominating function if $f(N(v))=\sum_{u\in N(v)}f(u)\ge w_i$ for every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be $\omega(f)=\sum_{v\in V(G)} f(v)$. The $w$-domination number of $G$, denoted by $\gamma_{w}(G)$, is the minimum weight among all $w$-dominating functions on $G$. If we impose restrictions on the minimum degree of $G$ when needed, under this approach we can define, for instance, the domination number, the total domination number, the $k$-domination number, the $k$-tuple domination number, the $k$-tuple total domination number, the Italian domination number, the total Italian domination number, and the $\{k\}$-domination number. Specifically, we show that $\gamma_{I}(G\circ H)=\gamma_{w}(G)$, where $w\in \{2\}\times\{0,1,2\}^{l}$ and $l\in \{2,3\}$. The decision on whether the equality holds for specific values of $w_0,\dots,w_l$ will depend on the value of the domination number of $H$. This paper also provides preliminary results on $\gamma_{w}(G)$ and raises the challenge of conducting a detailed study of the topic.