  Borodin and Kostochka proved that for $d_2 \geq 2d_1+2$ and a graph $G$ where
every subgraph $H$ satisfies $$ e(H) < \left(2 -
\frac{d_2+2}{(d_1+2)(d_2+1)}\right)n(H) + \frac{1}{d_2+1} $$ has a vertex
partition $V(G) = V_1 \cup V_2$ such that $G[V_i]$ has maximum degree at most
$d_i$ for each $i$. We show that under the same conditions we can additionally
conclude that each $G[V_i]$ is a forest.
