For any graph $G$ and any set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$ does not contain a copy of a graph in $\mathcal{F}$. Thus, $\iota(G,\{K_1\})$ is the domination number of $G$. For any integer $k \geq 1$, let $\mathcal{F}_{0,k} = \{K_{1,k}\}$, let $\mathcal{F}_{1,k}$ be the set of regular graphs of degree at least $k-1$, let $\mathcal{F}_{2,k}$ be the set of graphs whose chromatic number is at least $k$, and let $\mathcal{F}_{3,k}$ be the union of $\mathcal{F}_{0,k}$, $\mathcal{F}_{1,k}$ and $\mathcal{F}_{2,k}$. We prove that if $G$ is a connected $n$-vertex graph and $\mathcal{F} = \mathcal{F}_{0,k} \cup \mathcal{F}_{1,k}$, then $\iota(G, \mathcal{F}) \leq \frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle. This generalizes a bound of Caro and Hansberg on the $\{K_{1,k}\}$-isolation number, a bound of the author on the cycle isolation number, and a bound of Fenech, Kaemawichanurat and the author on the $k$-clique isolation number. By Brooks' Theorem, the same holds if $\mathcal{F} = \mathcal{F}_{3,k}$. The bounds are sharp.

翻译：对于任意图$G$和任意图族$\mathcal{F}$，令$\iota(G,\mathcal{F})$表示$G$中满足以下条件的最小顶点集$D$的大小：从$G$中删除$D$的闭邻域后得到的图不包含$\mathcal{F}$中任意图的副本。因此，$\iota(G,\{K_1\})$即为$G$的控制数。对于任意整数$k \geq 1$，令$\mathcal{F}_{0,k} = \{K_{1,k}\}$，$\mathcal{F}_{1,k}$为所有度数至少为$k-1$的正则图构成的集合，$\mathcal{F}_{2,k}$为所有色数至少为$k$的图构成的集合，而$\mathcal{F}_{3,k}$为$\mathcal{F}_{0,k}$、$\mathcal{F}_{1,k}$与$\mathcal{F}_{2,k}$的并集。我们证明：若$G$是连通的$n$顶点图且$\mathcal{F} = \mathcal{F}_{0,k} \cup \mathcal{F}_{1,k}$，则$\iota(G, \mathcal{F}) \leq \frac{n}{k+1}$，除非$G$是$k$-团，或$k=2$且$G$是$5$-圈。这一结果推广了Caro与Hansberg关于$\{K_{1,k}\}$-孤立数的界、作者关于圈孤立数的界，以及Fenech、Kaemawichanurat与作者关于$k$-团孤立数的界。根据Brooks定理，当$\mathcal{F} = \mathcal{F}_{3,k}$时结论同样成立。上述界是紧的。