Perfect Roman domination in middle graphs

TL;DR

This paper characterizes middle graphs where the Roman domination number equals the perfect Roman domination number, providing insights into their structural properties and the conditions for this equality.

Contribution

It offers a characterization of middle graphs with equal Roman and perfect Roman domination numbers, advancing understanding of domination parameters in graph theory.

Findings

Identifies conditions for equality of Roman and perfect Roman domination numbers in middle graphs

Provides structural characterization of such graphs

Enhances theoretical understanding of domination parameters

Abstract

The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which $f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on $G$. In this paper, we give a characterization of middle graphs with equal Roman domination and perfect Roman domination numbers.