Critical properties on Roman domination graphs

TL;DR

This paper investigates the properties of Roman domination in graphs, focusing on critical graphs where the Roman domination number decreases upon removal of vertices or edges, and fully characterizes those with a specific domination number of 4.

Contribution

It provides new results on Roman critical graphs and offers a complete characterization of critical graphs with Roman domination number equal to 4.

Findings

Characterization of Roman critical graphs with b3_R(G)=4

New results on the behavior of Roman domination number under graph modifications

Insights into the structure of graphs with minimal Roman domination

Abstract

A Roman domination function on a graph G is a function $r:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $r(u)=0$ is adjacent to at least one vertex $v$ for which $r(v)=2$. The weight of a Roman function is the value $r(V(G))=\sum_{u\in V(G)}r(u)$. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum weight of a Roman domination function on $G$. "Roman Criticality" has been defined in general as the study of graphs where the Roman domination number decreases when removing an edge or a vertex of the graph. In this paper we give further results in this topic as well as the complete characterization of critical graphs that have Toman Domination number $\gamma_R(G)=4$.