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Partitioning a triangle-free planar graph into a forest and a forest of bounded degree. (English) Zbl 1369.05169
Summary: 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.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory 05C07 Vertex degrees
##### Keywords:
$$(\mathcal{F},\mathcal{F}_5)$$-partition
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##### References:
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