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An $({\cal F},{\cal F}_d)$-partition of a graph is a vertex-partition intotwo sets $F$ and $F_d$ such that the graph induced by $F$ is a forest and theone induced by $F_d$ is a forest with maximum degree at most $d$. We prove thatevery triangle-free planar graph admits an $({\cal F},{\cal F}_5)$-partition.Moreover we show that if for some integer $d$ there exists a triangle-freeplanar graph that does not admit an $({\cal F},{\cal F}_d)$-partition, then itis an NP-complete problem to decide whether a triangle-free planar graph admitssuch a partition.

Partitioning a triangle-free planar graph into a forest and a forest of bounded degree