Total Roman domination in digraphs

Abstract

Let D be a finite and simple digraph with vertex set V (D). A Roman dominating function (RDF) on a digraph D is a function f : V (D) → {0, 1, 2} satisfying the condition that every vertex v with f (v) = 0 has an in-neighbor u with f (u) = 2. The weight of an RDF f is the value . An RDF f on D with no isolated vertex is called a total Roman dominating function if the subdigraph of D induced by the set {v ∈ V (D): f (v) ≠ 0} has no isolated vertex. The Roman domination number is the minimum weight of a total Roman dominating function on D. In this paper, we initiate the study of the total Roman domination number in digraphs and show its relationship to other domination parameters. In particular, we present some sharp bounds for the total Roman domination number and we determine the total Roman domination number of some classes of digraphs.