Italian Domination and Perfect Italian Domination on Sierpinski Graphs

TL;DR

This paper investigates the Italian domination and perfect Italian domination numbers of Sierpinski graphs, providing new insights into these graph parameters and their specific values for this class of fractal-like graphs.

Contribution

The paper determines the Italian domination number and perfect Italian domination number specifically for Sierpinski graphs, a class of fractal graphs, which was not previously known.

Findings

Calculated the Italian domination number for Sierpinski graphs.

Determined the perfect Italian domination number for Sierpinski graphs.

Provided formulas or bounds for these parameters in the context of Sierpinski graphs.

Abstract

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.