Impact of Vertex Addition on Italian Domination Number

TL;DR

This paper investigates how the corona operation and twin addition affect the Italian domination number in graphs, providing insights into graph domination properties under these modifications.

Contribution

It introduces the effects of the corona operator and twin addition on the Italian domination number, a novel analysis in graph theory.

Findings

Corona operation can increase or decrease the Italian domination number.

Adding twins has a predictable impact on the Italian domination number.

Results provide bounds and formulas for these graph modifications.

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(G) $ 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(G)}f(v) $ and Italian domination number, $ \gamma_I(G) $ is the minimum weight of an IDF. In this paper, we study the impact of corona operator and addition of twins on Italian domination number.