Partition of Sparse Graphs into Two Forests with Bounded Degree

Borodin and Kostochka proved that for d₂ ≥ 2d₁+2 and a graph G where every subgraph H satisfies e(H) < (2 - d₂+2/(d₁+2)(d₂+1))n(H) + 1/d₂+1 has a vertex partition V(G) = V₁ ∪ V₂ such that G[Vᵢ] has maximum degree at most dᵢ for each i. We show that under the same conditions we can additionally conclude that each G[Vᵢ] is a forest.