Mixed double Roman domination in graphs

Let G=(V,E) be a simple graph with vertex set V and edge set E. A mixed double Roman dominating function (MDRDF) of G is a function f : V boolean OR E -> {0,1,2,3} satisfying (1) for any element x is an element of V boolean OR E with f(x)=0, there must be either element y is an element of V boolean OR E, with f(y)=3, which is adjacent or incident to x, or either two elements y,z is an element of V boolean OR E, with f(y),f(z)=2 which are adjacent or incident to x; (2) for any element x is an element of V boolean OR E with f(x)=1, there must be either element y is an element of V boolean OR E, with f(y)>= 2, which is adjacent or incident to x. The weight of an MDRDF f is w(f)=f(V boolean OR E)=& sum;(x is an element of V boolean OR E )f(x) and the minimum weight among all the MDRD functions the MDRD-number, gamma(& lowast;)(dR)(G), of the graph G. In this paper we start the study of this variation of the classic Roman domination problem by setting some basic results, giving exact values and sharp bounds of the MDRD number and we approach the study of the complexity of the decision problem associated to the MDR domination in graphs.