有向图的意大利束缚数和加强数
The Italian bondage and reinforcement numbers of digraphs
摘要：
有向图$D$上的一个\textit{Italian dominating function}被定义为一个函数$f : V(D) \rightarrow \{0, 1, 2\}$，使得每个顶点$v \in V(D)$都有至少两个内邻居被指定为$f$下的$1$或一个内邻居被指定为$f(w) = 2$ .意大利控制函数$f$的\textit{weight}是值$ω(f) = f(V(D)) = \sum_{u \in V(D)} f(u)$。有向图$D$的\textit{Italian domination number}，记作$γ_I(D)$，是$D$上所有意大利控制函数的权的最小值。一个有向图$D$的\textit{Italian bondage number}，记作$b_I(D)$，是$A(D)$的最小弧数，在$D$中除去这些弧会产生一个有$γ_I(D') > γ_I(D)$的有向图$D'$。一个有向图$D$的\textit{Italian reinforcement number}，记作$r_I(D)$，是将其添加到$D$中导致一个有向图$D'$与$γ_I(D') < γ_I(D)$的额外弧的最小数目。本文主要研究了有向图中的意大利束缚数和意大利加强数，并给出了$b_I(D)$和$r_I(D)$的一些界。我们还确定了几类有向图的意大利束缚数和加强数。
英文摘要：
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 $ω(f) = f(V(D)) = \sum_{u \in V(D)} f(u)$. The \textit{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 \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 $γ_I(D') > γ_I(D)$. 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 $γ_I(D') < γ_I(D)$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
