On the isolation number of graphs with minimum degree four

An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $\iota (G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all isolating sets of $G$. We provide a technique for proving upper bounds on this parameter for graphs with a given minimum degree. For example, we show that if $G$ has order~$n$ and minimum degree at least~$4$, then $\iota (G)\le 13n\mathrm{/}41$, and if $G$ is also triangle-free, then $\iota (G)\le 3n\mathrm{/}10$.