Independent Roman Domination Stable and Vertex-Critical Graphs

A Roman dominating function (RDF) on a graph $G$ is a function $f: V(G) \rightarrow \{0, 1, 2\}$ for which every vertex assigned 0 is adjacent to a vertex assigned 2. The weight of an RDF is the value $\omega (f) = \sum _{u \in V(G)}f(u)$ . The minimum weight of an RDF on a graph $G$ is called the Roman domination number of $G$ . An RDF $f$ is called an independent Roman dominating function (IRDF) if the set $\{v\in V\mid f(v)\ge 1\}$ is an independent set. The minimum weight of an IRDF on a graph $G$ is called the independent Roman domination number of $G$ and is denoted by $i_{R}(G)$ . A graph $G$ is independent Roman domination stable if the independent Roman domination number of $G$ does not change under removal of any vertex. A graph $G$ is said to be independent Roman domination vertex critical or $i_{R}$ -vertex critical, if for any vertex $v$ in $G$ , $i_{R}(G-v) < i_{R}(G)$ . In this paper, we characterize independent Roman domination stable trees and we establish upper bounds on the order of independent Roman stable graphs. Also, we investigate the properties of $i_{R}$ - vertex critical graphs. In particular, we present some families of $i_{R}$ -vertex critical graphs and we characterize $i_{R}$ -vertex critical block graphs.