Sparse vertex cutsets and the maximum degree

We show that every graph $G$ of maximum degree $\mathrm{\Delta }$ and sufficiently large order has a vertex cutset $S$ of order at most $\mathrm{\Delta }$ that induces a subgraph $G[S]$ of maximum degree at most $\mathrm{\Delta }-3$. For $\mathrm{\Delta }\in \{4,5\}$, we refine this result by considering also the average degree of $G[S]$. If $G$ has no $K_{r,r}$ subgraph, then we show the existence of a vertex cutset that induces a subgraph of maximum degree at most $(1-\frac{1}{\left(\genfrac{}{}{0px}{}{r}{2}\right)})\mathrm{\Delta }+O(1)$.