Independent Roman Domination Stable and Vertex-Critical Graphs

A Roman dominating function (RDF) on a graph G is a function f : V(G) -> {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 ) = Sigma(u is an element of 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 is an element of V vertical bar f (v) >= 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 - nu) < 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.