For any positive integer $k$ and any $n$-vertex graph $G$, let $\iota(G,k)$ 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 $D$ contains no $k$-clique. Thus, $\iota(G,1)$ is the domination number of $G$. We prove that if $G$ is connected, then $\iota(G,k) \leq \frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle. The bound is sharp. The case $k=1$ is a classical result of Ore, and the case $k=2$ is a recent result of Caro and Hansberg. Our result solves a problem of Caro and Hansberg.

中文翻译：

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k-cliques的隔离

对于任何正整数 $k$ 和任何 $n$-顶点图 $G$，让 $\iota(G,k)$ 表示 $G$ 顶点的最小集合 $D$ 的大小，使得图获得从 $G$ 删除 $D$ 的封闭邻域不包含 $k$-clique。因此，$\iota(G,1)$ 是$G$ 的支配数。我们证明如果 $G$ 是连通的，那么 $\iota(G,k) \leq \frac{n}{k+1}$ 除非 $G$ 是一个 $k$-clique 或 $k = 2$ 并且$G$ 是一个 $5$-周期。界限是尖锐的。案例$k=1$ 是Ore 的经典结果，案例$k=2$ 是Caro 和Hansberg 的最新结果。我们的结果解决了 Caro 和 Hansberg 的问题。