On the cycle isolation number of triangle-free graphs

For a graph G, a subset $S\subseteq V\left(G\right)$ is called a cycle isolating set of G if $G-N\left[D\right]$ contains no cycle. The cycle isolation number of G, denoted by ${\iota }_c\left(G\right)$, is the minimum cardinality of a cycle isolating set of G. Recently, Borg proved that if G is a connected n-vertex graph that is not a triangle, then ${\iota }_c\left(G\right)\le \frac{n}{4}$. In this paper, we prove that if G is a connected triangle-free n-vertex graph that is not a 4-cycle, then ${\iota }_c\left(G\right)\le \frac{n}{5}$. In particular, we characterize the subcubic graphs that attain the bound. For graphs with larger girth, several conjectures are proposed.