Given a set $\mathcal{F}$ of graphs, we call a copy of a graph in $\mathcal{F}$ an $\mathcal{F}$-graph. The $\mathcal{F}$-isolation number of a graph $G$, denoted by $\iota(G,\mathcal{F})$, is the size of a smallest subset $D$ of the vertex set $V(G)$ such that the closed neighbourhood of $D$ intersects the vertex sets of the $\mathcal{F}$-graphs contained by $G$ (equivalently, $G - N[D]$ contains no $\mathcal{F}$-graph). Thus, $\iota(G,\{K_1\})$ is the domination number of $G$. The second author showed that if $\mathcal{F}$ is the set of cycles and $G$ is a connected $n$-vertex graph that is not a triangle, then $\iota(G,\mathcal{F}) \leq \left \lfloor \frac{n}{4} \right \rfloor$. This bound is attainable for every $n$ and solved a problem of Caro and Hansberg. A question that arises immediately is how smaller an upper bound can be if $\mathcal{F} = \{C_k\}$ for some $k \geq 3$, where $C_k$ is a cycle of length $k$. The problem is to determine the smallest real number $c_k$ (if it exists) such that for some finite set $\mathcal{E}_k$ of graphs, $\iota(G, \{C_k\}) \leq c_k |V(G)|$ for every connected graph $G$ that is not an $\mathcal{E}_k$-graph. The above-mentioned result yields $c_3 = \frac{1}{4}$ and $\mathcal{E}_3 = \{C_3\}$. The second author also showed that if $k \geq 5$ and $c_k$ exists, then $c_k \geq \frac{2}{2k + 1}$. We prove that $c_4 = \frac{1}{5}$ and determine $\mathcal{E}_4$, which consists of three $4$-vertex graphs and six $9$-vertex graphs. The $9$-vertex graphs in $\mathcal{E}_4$ were fully determined by means of a computer program. A method that has the potential of yielding similar results is introduced.

翻译：给定一个图族 $\mathcal{F}$，我们将图族中的图称为 $\mathcal{F}$-图。图 $G$ 的 $\mathcal{F}$-孤立数，记为 $\iota(G,\mathcal{F})$，是顶点集 $V(G)$ 中最小子集 $D$ 的大小，使得 $D$ 的闭邻域与 $G$ 包含的所有 $\mathcal{F}$-图的顶点集相交（等价地，$G - N[D]$ 不含任何 $\mathcal{F}$-图）。因此，$\iota(G,\{K_1\})$ 是图 $G$ 的控制数。第二作者证明：若 $\mathcal{F}$ 是圈集且 $G$ 是连通 $n$ 顶点图且不是三角形，则 $\iota(G,\mathcal{F}) \leq \left \lfloor \frac{n}{4} \right \rfloor$。该界对每个 $n$ 均可达到，并解决了 Caro 与 Hansberg 的一个问题。一个随之而来的问题是：若 $\mathcal{F} = \{C_k\}$，其中 $k \geq 3$ 且 $C_k$ 是长度为 $k$ 的圈，则上界能小到何种程度。该问题在于确定最小实数 $c_k$（若存在），使得存在有限图集 $\mathcal{E}_k$，对每个不是 $\mathcal{E}_k$-图的连通图 $G$，有 $\iota(G, \{C_k\}) \leq c_k |V(G)|$。上述结果给出 $c_3 = \frac{1}{4}$ 且 $\mathcal{E}_3 = \{C_3\}$。第二作者还证明：若 $k \geq 5$ 且 $c_k$ 存在，则 $c_k \geq \frac{2}{2k + 1}$。我们证明 $c_4 = \frac{1}{5}$ 并确定 $\mathcal{E}_4$，它包含三个 $4$ 顶点图和六个 $9$ 顶点图。$\mathcal{E}_4$ 中的 $9$ 顶点图通过计算机程序完全确定。本文还介绍了一种有望得到类似结果的方法。