The Italian bondage and reinforcement numbers of digraphs

TL;DR

This paper introduces and studies the Italian bondage and reinforcement numbers in directed graphs, providing bounds and exact values for certain classes, expanding the understanding of domination concepts in digraphs.

Contribution

It initiates the study of Italian bondage and reinforcement numbers in digraphs, offering bounds and exact values for specific classes, a novel contribution in domination theory.

Findings

Established bounds for Italian bondage and reinforcement numbers.

Determined exact values for these numbers in some classes of digraphs.

Laid groundwork for future research in Italian domination in digraphs.

Abstract

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 $\gamma_I(D)$, 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 $b_I(D)$, is the minimum number of arcs of $A(D)$ whose removal in $D$ results in a digraph $D'$ with $\gamma_I(D') > \gamma_I(D)$. The \textit{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 $\gamma_I(D') < \gamma_I(D)$. 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.