Independent Roman Domination Stable and Vertex-Critical Graphs

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