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An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ \gamma_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f$ except for exactly one vertex $v$ for which $ f(v) = 2 $ or for exactly two vertices $v$ and $w$ for which $ f(v) = f(w) = 1 $. The weight of a PID- function is $f(V)=\sum_{u \in V(G)}f(u)$. The perfect Italian domination number of $G$, denoted by $ \gamma^{p}_{I}(G), $ is the minimum weight of a PID-function of $G$. In this paper we obtain the Italian domination number and perfect Italian domination number of Sierpi\'{n}ski graphs. |
