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