Roman domination in graphs: The class ℛUV R

For a graph G = (V,E), a Roman dominating function (RDF) f : V →{0, 1, 2} has the property that every vertex v ∈ V with f(v) = 0 has a neighbor u with f(u) = 2. The weight of a RDF f is the sum f(V ) = Σv∈Vf(v), and the minimum weight of a RDF on G is the Roman domination number γR(G) of G. The Roman bondage number bR(G) of G is the minimum cardinality of all sets F ⊆ E for which γR(G − F) > γR(G). A graph G is in the class ℛUV R if the Roman domination number remains unchanged when a vertex is deleted. In this paper, we obtain tight upper bounds for γR(G) and bR(G) provided a graph G is in ℛUV R. We present necessary and sufficient conditions for a tree to be in the class ℛUV R. We give a constructive characterization of ℛUV R-trees using labelings.