  A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by
subsequent removals of vertices of degree $k$ or less. We prove that every
connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced
subgraph containing at least $(38-d)/36$ of its vertices. This shows that every
planar graph of order $n$ has a 4-degenerate induced subgraph of order more
than $8/9 \cdot n$. We also consider a local variation of this problem and show
that in every planar graph with at least 7 vertices, deleting a suitable vertex
allows us to subsequently remove at least 6 more vertices of degree four or
less.
