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