Perfect Italian Domination Number of Graphs

TL;DR

This paper establishes bounds and exact values for the perfect Italian domination number in graph theory, explores its relationships with other domination parameters, and addresses realization problems and open questions.

Contribution

It introduces new bounds, exact values, and relationships for the perfect Italian domination number, expanding understanding of this parameter in graph theory.

Findings

Upper bounds for Cartesian product graphs' perfect Italian domination number

Exact values for specific graph classes

Relationships with Roman and perfect domination numbers

Abstract

In this paper, an upper bound for the perfect Italian domination number of the cartesian product of any two graphs is obtained and the exact value of this parameter for cartesian product of some special graphs are obtained. We have also proved that for any two positive integers $a$, $b$ there exists a graph $G$ and an induced subgraph $H$ of $G$ such that $\gamma_I^p(G) = a$ and $\gamma_I^p(H) = b$. Relationship of the perfect Italian domination number with the Roman domination number and the perfect domination number of a graph $G$ are obtained and the corresponding realization problems are also solved. We have also obtained the perfect Italian domination number of the Mycielskian of a graph in terms of the perfect domination number of the graph. Some open problems related to this parameters are also included.