Some progress on the mixed roman domination in graphs

Let G = (V, E) be a simple graph with vertex setxs V and edge set E. A mixed Roman dominating function of G is a function f : V ∪ E → {0, 1, 2} satisfying the condition that every element x ∈ V ∪ E for which f(x) = 0 is adjacent or incident to at least one element y ∈ V ∪ E for which f(y) = 2. The weight of a mixed Roman dominating function f is ω(f) = ∑x∈V∪E f(x). The mixed Roman domination number $ \gamma{^*_R}(G) $ of G is the minimum weight of a mixed Roman dominating function of G. We first show that the problem of computing $ \gamma{^*_R}(G) $ is NP-complete for bipartite graphs and then we present upper and lower bounds on the mixed Roman domination number, some of them are for the class of trees