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Abstract
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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 DRDF f is the value f (V) = 6u∈V(G)
f (u). The total
double Roman dominating function (TDRDF) on a graph G without isolated vertices is a DRDF f 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 γtdR(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
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