Isolation of connected graphs

Abstract

For a connected $n$-vertex graph $G$ and a set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$ contains no graph in $\mathcal{F}$. Let $\mathcal{E}_k$ denote the set of connected graphs that have at least $k$ edges. By a result of Caro and Hansberg, $\iota(G,\mathcal{E}_1) \leq n/3$ if $n \neq 2$ and $G$ is not a $5$-cycle. The author recently showed that if $G$ is not a triangle and $\mathcal{C}$ is the set of cycles, then $\iota(G,\mathcal{C}) \leq n/4$. We improve this result by showing that $\iota(G,\mathcal{E}_3) \leq n/4$ if $G$ is neither a triangle nor a $7$-cycle. Let $r$ be the number of vertices of $G$ that have only one neighbour. We determine a set $\mathcal{S}$ of six graphs such that $\iota(G,\mathcal{E}_2) \leq (4n - r)/14$ if $G$ is not a copy of a member of $\mathcal{S}$. The bounds are sharp.