Let G be a simple graph with vertex set V . A double Roman dominating function (DRDF) on G is a function f : V ! f0; 1; 2; 3g satisfying that if f(v) = 0, then the vertex v must be adjacent to at least two vertices assigned 2 or one vertex assigned 3 under f, whereas if f(v) = 1, then the vertex v must be adjacent to at least one vertex assigned 2 or 3. The weight of a DRDF f is the sum P v2V f(v). A total double Roman dominating function (TDRDF) on a graph G with no isolated vertex is a DRDF f on G with the additional property that the subgraph of G induced by the set fv 2 V : f(v) 6= 0g has no isolated vertices. The total double Roman domination number tdR(G) is the minimum weight of a TDRDF on G. In this paper, we give several relations between the total double Roman domination number of a graph and other domination parameters and we determine the total double Roman domination number of some classes of graphs.
