For any set $\mathcal{F}$ of graphs and any graph $G$, 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 $\mathcal{F}_{1,k}$ 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 $\mathcal{F}_{1,k}$ and $\mathcal{F}_{2,k}$. We prove that for each $i \in \{1, 2, 3\}$, if $G$ is a connected $m$-edge graph that is not a $k$-clique, then $\iota(G, \mathcal{F}_{i,k}) \leq \frac{m+1}{{k \choose 2} + 2}$. We also determine the graphs for which the bound is attained. Among the consequences are a sharp bound of Fenech, Kaemawichanurat and the present author on the $k$-clique isolation number and a sharp bound on the cycle isolation number.

翻译：对于任意图族$\mathcal{F}$和图$G$，定义$\iota(G,\mathcal{F})$为$G$中最小顶点集$D$的大小，使得从$G$中删除$D$的闭邻域后得到的图不包含$\mathcal{F}$中任何图的副本。因此，$\iota(G,\{K_1\})$是$G$的支配数。对于任意整数$k \geq 1$，设$\mathcal{F}_{1,k}$为度数至少为$k-1$的正则图族，$\mathcal{F}_{2,k}$为色数至少为$k$的图族，$\mathcal{F}_{3,k}$为$\mathcal{F}_{1,k}$与$\mathcal{F}_{2,k}$的并集。我们证明：对于每个$i \in \{1, 2, 3\}$，若$G$是非$k$-团的连通$m$边图，则$\iota(G, \mathcal{F}_{i,k}) \leq \frac{m+1}{{k \choose 2} + 2}$。此外，我们确定了达到该边界的图。该结果的一个推论包括Fenech、Kaemawichanurat与本文作者关于$k$-团隔离数的精确界，以及关于圈隔离数的精确界。