The Italian bondage and reinforcement numbers of digraphs

An \textit{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow {0, 1, 2}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. The \textit{weight} of an Italian dominating function $f$ is the value $\omega(f) = f(V(D)) = \sum{u \in V(D)} f(u)$. The \textit{Italian domination number} of a digraph $D$, denoted by $\gammaI(D)$, is the minimum taken over the weights of all Italian dominating functions on $D$. The \textit{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 $\gammaI(D') > \gammaI(D)$. The \textit{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 $\gammaI(D') < \gammaI(D)$. 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.