Roman domination: changing, unchanging, $\gamma_R$-graphs

TL;DR

This paper explores the properties and classifications of Roman domination in graphs, analyzing how the Roman domination number changes with modifications, and introduces the concept of $oldsymbol{oldsymbol{oldsymbol{ ext{}}}}$-graphs based on minimal Roman dominating functions.

Contribution

It introduces a new framework for understanding Roman domination classes, studies properties of Roman domination critical graphs, and initiates the study of $oldsymbol{oldsymbol{oldsymbol{ ext{}}}}$-graphs formed from minimal Roman dominating functions.

Findings

Identified relationships among six classes of graphs based on Roman domination number changes.

Defined and studied properties of Roman domination $k$-critical graphs.

Initiated the study of $oldsymbol{oldsymbol{oldsymbol{ ext{}}}}$-graphs and their adjacency relations.

Abstract

A Roman dominating function (RD-function) on a graph $G = (V(G), E(G))$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight $f(V(G))$ of a RD-function $f$ on $G$ is the value $\Sigma_{v\in V(G)} f (v)$. The {\em Roman domination number} $\gamma_{R}(G)$ of $G$ is the minimum weight of a RD-function on $G$. The six classes of graphs resulting from the changing or unchanging of the Roman domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. We consider relationships among the classes, which are illustrated in a Venn diagram. A graph $G$ is Roman domination $k$-critical if the removal of any set of $k$ vertices decreases the Roman domination number. Some initial properties of these graphs are studied. The $\gamma_R$-graph of a graph $G$ is any graph which vertex set is the collection $\mathscr{D}_R(G)$ of all minimum weight RD-functions on $G$. We define adjacency between any two elements of $\mathscr{D}_R(G)$ in several ways, and initiate the study of the obtained $\gamma_R$-graphs.