Let G= (V, E) be a simple graph with vertex set V and edge set E. A mixed Roman dominating function (MRDF) of G is a function f: V∪ E→ { 0 , 1 , 2 } satisfying the condition 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 MRDF f is ω(f) = ∑ x∈V∪Ef(x). The mixed Roman domination number of G is the minimum weight of a mixed Roman dominating function of G. In this paper, we initiate the study of the mixed Roman domination number and we present bounds for this parameter. We characterize the graphs attaining an upper bound and the graphs having small mixed Roman domination numbers.

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect match- ing between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V(G) → {0,1,2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V ) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. Our main results show that γR(GG) takes on a limited number of values in terms of the domination number of GG and the Roman domination numbers of G and G.