Sparse vertex cutsets and the maximum degree

We show that every graph $G$ of maximum degree $\Delta$ and sufficiently large order has a vertex cutset $S$ of order at most $\Delta$ that induces a subgraph $G[S]$ of maximum degree at most $\Delta-3$. For $\Delta\in \{ 4,5\}$, 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 $\left(1-\frac{1}{{r\choose 2}}\right)\Delta+O(1)$.