A set D of vertices in a graph G is a cycle isolating set of G if \(G-N[D]\) contains no cycle. The cycle isolation number of G, denoted by \(\iota _c(G)\) , is the minimum cardinality of a cycle isolating set of G. In this paper, we prove that if G is a connected graph of size m that is not a \(C_3\) , then \(\iota _c(G) \le \frac{m+1}{5}\) , and we characterize the extremal graphs. Moreover, we conjecture that if G is a connected graph of size m that is not a \(C_g\) , then \(\iota _c(G) \le \frac{m+1}{g+2}\) , where g is the girth of G.
