Italian Domination and Perfect Italian Domination on Sierpinski Graphs

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.