Roman domination in graphs: the class $\mathcal{R}_\mathbf{UV R}$

TL;DR

This paper studies Roman domination in graphs, focusing on the class where the Roman domination number is unaffected by vertex deletion, providing bounds, characterizations, and constructive methods for such graphs and trees.

Contribution

It introduces the class R_{UVR} of graphs with stable Roman domination number under vertex removal, and offers bounds, conditions, and a labelling-based characterization for these graphs and trees.

Findings

Established tight upper bounds for  gamma_R(G) and  b_R(G) in R_{UVR} graphs.

Derived necessary and sufficient conditions for trees to belong to R_{UVR}.

Developed a constructive labelling method to characterize R_{UVR} trees.

Abstract

For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is the sum $f (V) = \cup_{v\in V} f (v)$, and the minimum weight of a Roman dominating function on $G$ is the Roman domination number $\gamma_R(G)$ of $G$. The Roman bondage number $b_R(G)$ of $G$ is the minimum cardinality of all sets $F \subseteq E$ for which $\gamma_R(G - F) > \gamma_R(G)$. A graph $G$ is in the class $\mathcal{R}_{UVR}$ if the Roman domination number remains unchanged when a vertex is deleted. In this paper we obtain tight upper bounds for $\gamma_R(G)$ and $b_R(G)$ provided a graph $G$ is in $\mathcal{R}_{UVR}$. We present necessary and sufficient conditions for a tree to be in the class $\mathcal{R}_{UV R}$. We give a constructive characterization of $\mathcal{R}_{UVR}$-trees using labellings.