Summary: 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 \(G\). P. Borg [Graphs Comb. 36, No. 3, 631–637 (2020; Zbl 1439.05117)] showed that if \(\mathcal{F}\) is the set of cycles and \(G\) is a connected \(n\)-vertex graph that is not a triangle, then \(\iota(G, \mathcal{F}) \leq \lfloor \frac{ n}{ 4} \rfloor \). This bound is attainable for every \(n\) and solved a problem of Y. Caro and A. Hansberg [Filomat 31, No. 12, 3925–3944 (2017; Zbl 1488.05367)]. A question that arises immediately is how much 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 \(c_k \geq \frac{ 2}{ 2 k + 1} \). 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.