On the Total Double Roman Domination

Let G = (V, E) be a simple graph. A double Roman dominating function (DRDF) on G is a function f from the vertex set V of G into {0, 1, 2, 3} such that if f (u) = 0, then u must have at least two neighbors assigned 2 or one neighbor assigned 3 under f , and if f (u) = 1, then u must have at least one neighbor assigned at least 2 under f . The weight of a DRDFf is the value f (V) = Σu∈V(G)f (u). The total double Roman dominating function (TDRDF) on a graph G without isolated vertices is a DRDFf on G with the additional condition that the subgraph of G induced by the set {v ∈ V : f (v) ≥ 1} is isolated-free. The total double Roman domination number ytdR(G) is the smallest weight among all TDRDFs on G. In this paper, we first show that the decision problem for the total double Roman domination is NP-hard for chordal and bipartite graphs, and then we establish some sharp bounds on total double Roman domination number.