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

TL;DR

This paper proves that every triangle-free planar graph can be partitioned into a forest and a forest with maximum degree five, and shows that deciding such partitions for other degrees is NP-complete.

Contribution

The authors establish the existence of an ({ m F},{ m F}_5)-partition for all triangle-free planar graphs and prove the NP-completeness of deciding partitions for other degrees.

Findings

Every triangle-free planar graph admits an ({ m F},{ m F}_5)-partition.

Deciding ({ m F},{ m F}_d)-partitions for some degrees is NP-complete.

Partitioning problems become computationally hard for certain degrees.

Abstract

An $({\cal F},{\cal F}_d)$-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 $({\cal F},{\cal F}_5)$-partition. Moreover we show that if for some integer $d$ there exists a triangle-free planar graph that does not admit an $({\cal F},{\cal F}_d)$-partition, then it is an NP-complete problem to decide whether a triangle-free planar graph admits such a partition.