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- W3161886405 abstract "Let $w=(w_0,w_1, dots,w_l)$ be a vector of nonnegative integers such that $ w_0ge 1$. Let $G$ be a graph and $N(v)$ the open neighbourhood of $vin V(G)$. We say that a function $f: V(G)longrightarrow {0,1,dots ,l}$ is a $w$-dominating function if $f(N(v))=sum_{uin N(v)}f(u)ge w_i$ for every vertex $v$ with $f(v)=i$. The weight of $f$ is defined to be $omega(f)=sum_{vin V(G)} f(v)$. Given a $w$-dominating function $f$ and any pair of adjacent vertices $v, uin V(G)$ with $f(v)=0$ and $f(u)>0$, the function $f_{urightarrow v}$ is defined by $f_{urightarrow v}(v)=1$, $f_{urightarrow v}(u)=f(u)-1$ and $f_{urightarrow v}(x)=f(x)$ for every $xin V(G)setminus{u,v}$. We say that a $w$-dominating function $f$ is a secure $w$-dominating function if for every $v$ with $f(v)=0$, there exists $uin N(v)$ such that $f(u)>0$ and $f_{urightarrow v}$ 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 $Gcirc H$ 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 $gamma_{(1,1)}^s(H)$." @default.
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- W3161886405 date "2021-05-11" @default.
- W3161886405 modified "2023-09-27" @default.
- W3161886405 title "From (secure) w-domination in graphs to protection of lexicographic product graphs" @default.
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