The Italian bondage and reinforcement numbers of digraphs
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An Italian dominating function on a digraph D with vertex set V(D) is defined as a function f : V(D) →{0, 1, 2} such that every vertex v ∈ V(D) with f(v) = 0 has at least two in-neighbors assigned 1 under f or one in-neighbor w with f(w) = 2. The weight of an Italian dominating function f is the value ω(f) = f(V(D)) = ∑_u ∈ V(D) f(u). The Italian domination number of a digraph D, denoted by γ_I(D), is the minimum taken over the weights of all Italian dominating functions on D. The Italian bondage number of a digraph D, denoted by b_I(D), is the minimum number of arcs of A(D) whose removal in D results in a digraph D' with γ_I(D') > γ_I(D). The Italian reinforcement number of a digraph D, denoted by r_I(D), is the minimum number of extra arcs whose addition to D results in a digraph D' with γ_I(D') < γ_I(D). In this paper, we initiate the study of Italian bondage and reinforcement numbers in digraphs and present some bounds for b_I(D) and r_I(D). We also determine the Italian bondage and reinforcement numbers of some classes of digraphs.
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