STABILITY OF WEAK ROMAN DOMINATION UPON VERTEX DELETION

Let G= (V, E) be a graph and f : V→{0,1,2}be a function. We write f = (V0, V1, V2), where Vi = {v|f(v) = i},I = 0,1,2. A vertex inV0 is said to be defended with respect to the function f, if it is adjacent to a vertex inV1 ∪ V2. A vertex that fails to satisfy this condition is said to be undefended with respect to f. The function f is said to be a weak Roman dominating function (WRDF) if for each vertex u ∈ V0, there exists a vertex u ∈ V1∪ V2, such that under the new function f′ defined on V by f′ (u) = 1,f′(v) =f(v)−1 and f′ (w) = f (w) for all vertices in V\{u, v}, no vertex in V is undefended. The weight of the WRDF f = (V0, V1, V2) is |V1|+ 2|V2|. The minimum weight of a WRDF defined on V is called the weak Roman domination number of G and is denoted by γr (G). Two classes of graphs, rUVR, consisting of those graphs in which for any vertex v ∈ V(G), γr (G−v) = γr (G) and rCVR, consisting of those graphs in which for any v ∈ V(G), γr (G−v)̸=γr(G), assume importance. In this paper we characterize certain graphs for membership in these classes.