Total Double Roman Domination in Graphs

Let g be a simple graph with vertex set v. a double roman dominating function (drdf) on g is a function f:v→{0,1,2,3} 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 ∑v∈vf(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 {v∈v:f(v)≠0} has no isolated vertices. the total double roman domination number γtdr(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.