Characterization of Roman domination critical unicyclic graphs

A Roman dominating function on a graph G is a function f : V(G) -> {0, 1, 2} satisfying the condition that every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the value f(V(G)) = Sigma(u is an element of V(G)) f(u). The Roman domination number, gamma(R)(G), of G is the minimum weight of a Roman dominating function on G. A graph G is said to be Roman domination vertex critical or just gamma(R)-vertex critical, if gamma(R)(G - v) < gamma(R)(G) for any vertex v is an element of V(G). Similarly, G is Roman domination edge critical or just gamma(R)-edge critical, if gamma(R)(G + e) < gamma(R)(G) for any edge e is not an element of E(G). In this paper, we characterize gamma(R)-vertex critical connected unicyclic graphs as well gamma(R)-edge critical connected unicyclic graphs.