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

Abstract

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

16 pages, 12 figures