Let D=(V,A)D=(V,A) be a finite and simple digraph. A Roman dominating function on DD is a labeling f:V(D)→{0,1,2}f:V(D)→{0,1,2} such that every vertex with label 0 has an in-neighbor with label 2. The weight of an RDF ff is the value ω(f)=∑v∈Vf(v)ω(f)=∑v∈Vf(v). The minimum weight of a Roman dominating function on a digraph DD is called the Roman domination number, denoted by γR(D)γR(D). The Roman bondage number bR(D)bR(D) of a digraph DD with maximum out-degree at least two is the minimum cardinality of all sets A′⊆AA′⊆A for which γR(D−A′)>γR(D)γR(D−A′)>γR(D). In this paper, we initiate the study of the Roman bondage number of a digraph. We determine the Roman bondage number in several classes of digraphs and give some sharp bounds.
