From (secure) w-domination in graphs to protection of lexicographic product graphs
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Let w=(w_0,w_1, …,w_l) be a vector of nonnegative integers such that w_0≥ 1. Let G be a graph and N(v) the open neighbourhood of v∈ V(G). We say that a function f: V(G)⟶{0,1,… ,l} is a w-dominating function if f(N(v))=∑_u∈ N(v)f(u)≥ w_i for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=∑_v∈ V(G) f(v). Given a w-dominating function f and any pair of adjacent vertices v, u∈ V(G) with f(v)=0 and f(u)>0, the function f_u→ v is defined by f_u→ v(v)=1, f_u→ v(u)=f(u)-1 and f_u→ v(x)=f(x) for every x∈ V(G)∖{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∈ N(v) such that f(u)>0 and f_u→ v is a w-dominating function as well. The (secure) w-domination number of G, denoted by (γ_w^s(G)) γ_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∘ H are related to γ_w^s(G) or γ_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 γ_(1,0)^s(H), while in the case of the total version of these parameters, the decision will depend on the value of γ_(1,1)^s(H).
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