Roman $\{2\}$-domination in graphs and graph products

TL;DR

This paper classifies graphs based on their Roman {2}-domination number and explores how this number behaves under various graph operations, providing a comprehensive understanding of this graph invariant.

Contribution

It classifies all graphs with specific Roman {2}-domination numbers and analyzes the effect of graph operations on this parameter, advancing the theoretical understanding of Roman domination.

Findings

Classified graphs with Roman {2}-domination number in {2,3,4,n-2,n-1,n}

Derived results on Roman {2}-domination number for various graph operations

Provided structural insights into Roman {2}-domination in graphs

Abstract

For a graph $G=(V,E)$ of order $n$, a Roman $\{2\}$-dominating function $f:V\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$, or $v$ is adjacent to least two vertices assigned $1$ under $f$. In this paper, we classify all graphs with Roman $\{2\}$-domination number belonging to the set $\{2,3,4,n-2,n-1,n\}$. Furthermore, we obtain some results about Roman $\{2\}$-domination number of some graph operations.