From (secure) w-domination in graphs to protection of lexicographic product graphs

Let $w=(w_0,w_1, \dots,w_l)$ be a vector of nonnegative integers such that $ w_0\ge 1$. Let $G$ be a graph and $N(v)$ the open neighbourhood of $v\in V(G)$. 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 $\omega(f)=\sum_{v\in V(G)} f(v)$. Given a $w$-dominating function $f$ and any pair of adjacent vertices $v, u\in V(G)$ with $f(v)=0$ and $f(u)>0$, the function $f_{u\rightarrow v}$ is defined by $f_{u\rightarrow v}(v)=1$, $f_{u\rightarrow v}(u)=f(u)-1$ and $f_{u\rightarrow v}(x)=f(x)$ for every $x\in V(G)\setminus\{u,v\}$. We say that a $w$-dominating function $f$ is a secure $w$-dominating function if for every $v$ with $f(v)=0$, there exists $u\in N(v)$ such that $f(u)>0$ and $f_{u\rightarrow v}$ is a $w$-dominating function as well. The (secure) $w$-domination number of $G$, denoted by ($\gamma_{w}^s(G)$) $\gamma_{w}(G)$, is defined as the minimum weight among all (secure) $w$-dominating functions. In this paper, we show how the secure (total) domination number and the (total) weak Roman domination number of lexicographic product graphs $G\circ H$ are related to $\gamma_w^s(G)$ or $\gamma_w(G)$. For the case of the secure domination number and the weak Roman domination number, the decision on whether $w$ takes specific components will depend on the value of $\gamma_{(1,0)}^s(H)$, while in the case of the total version of these parameters, the decision will depend on the value of $\gamma_{(1,1)}^s(H)$.