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