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Abstract
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Let D be a finite and simple digraph with vertex set V (D). A double Roman dominating
function (DRDF) on digraph D is a function f : V (D) → {0, 1, 2, 3} such that every
vertex with label 0 has an in-neighbor with label 3 or two in-neighbors with label 2
and every vertex with label 1 have at least one in-neighbor with label at least 2. The
weight of a DRDF f is the value ω(f) = Pv∈V (D) f(v). A DRDF f on D with no
isolated vertex is called a total double Roman dominating function if the subgraph of
D induced by the set {v ∈ V (D) | f(v) = 0} has no isolated vertex. In this paper, we
initiate the study of the total double Roman domination number in digraphs and show its
relationship to other domination parameters. In particular, we present some bounds for
the total double Roman domination number and we determine the total double Roman
domination number of some classes of digraphs.
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