The Italian bondage and reinforcement numbers of digraphs - Kim, Kijung (Electronic Journal of Graph Theory and Applications (EJGTA), 2025)
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 γI(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 bI(D), is the minimum number of arcs of A(D) whose removal in D results in a digraph D′ with γI(D′) > γI(D). The Italian reinforcement number of a digraph D, denoted by rI(D), is the minimum number of extra arcs whose addition to D results in a digraph D′ with γI(D′) < γI(D). In this paper, we initiate the study of Italian bondage and reinforcement numbers in digraphs and present some bounds for bI(D) and rI(D). We also determine the Italian bondage and reinforcement numbers of some classes of digraphs.