From (Secure) w-Domination in Graphs to Protection of Lexicographic Product Graphs

Abstract

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 $\gamma _{(1,1)}^s(H)$.