New results on the 1-isolation number of graphs without short cycles

Let $G$ be a graph. A subset $D\subseteq V\left(G\right)$ is called a 1-isolating set of $G$ if $\Delta (G-N\left[D\right])\le 1$, that is, $G-N\left[D\right]$ consists of isolated edges and isolated vertices only. The 1-isolation number of $G$, denoted by ${\iota }_1\left(G\right)$, is the cardinality of a smallest 1-isolating set of $G$. In this paper, we prove that if $G\notin \{P_3,C_3,C_7,C_{11}\}$ is a connected graph of order $n$ without 6-cycles, or without induced 5- and 6-cycles, then ${\iota }_1\left(G\right)\le \frac{n}{4}$. Both bounds are sharp.