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

让\(w=(w_0,w_1, \dots ,w_l)\)是一个非负整数向量，使得\( w_0\ge 1\)。设G是一个图，N ( v ) 是\(v\in V(G)\)的开邻域。我们说一个函数\（F：V（G）\ longrightarrow \ {0,1，\点，升\} \）是一个瓦特-dominating功能如果\（F（N（V））= \总和_ { u\in N(v)}f(u)\ge w_i\)对于每个顶点v和\(f(v)=i\)。f的权重定义为\(\omega (f)=\sum _{v\in V(G)} f(v)\)。给定一个w支配函数f和任何一对相邻的顶点\(v, u\in V(G)\)与\(f(v)=0\)和\(f(u)>0\)，函数\(f_{u\ rightarrow v}\)由\(f_{u\rightarrow v}(v)=1\)、\(f_{u\rightarrow v}(u)=f(u)-1\)和\(f_ {u\rightarrow v}(x)=f(x)\)对于每个\(x\in V(G){\setminus }\{u,v\}\)。我们说一个W¯¯ -dominating功能˚F是一个安全的w ^如果-dominating功能为每v与\（F（V）= 0 \） ，存在\（U \在N（V）\） ，使得\（F (u)>0\)和\(f_{u\rightarrow v}\)也是一个w支配函数。G的（安全）w支配数，由 ( \(\gamma _{w}^s(G)\) ) \(\gamma _{w}(G)\) 表示，被定义为最小权重在所有（安全）w主导功能中。在本文中，我们展示了词典产品图\（G \ circ H \）的安全（总）支配数和（总）弱罗马支配数与\（\ gamma _w ^ s（G）\）之间的关系。或\(\gamma _w(G)\)。对于安全统治数和弱罗马统治数的情况，决定w采取特定组件将取决于\(\gamma _{(1,0)}^s(H)\) 的值，而在这些参数的总版本的情况下，决定将取决于\的值(\gamma _{(1,1)}^s(H)\)。