Partitioning sparse graphs into an independent set and a forest of bounded degree

TL;DR

This paper establishes conditions based on maximum average degree for partitioning sparse graphs into an independent set and a bounded degree forest, advancing understanding of graph decompositions.

Contribution

It provides new bounds on maximum average degree ensuring such partitions exist, extending previous results in graph theory.

Findings

Partitions exist for graphs with max average degree less than M under specified bounds.

New bounds relate maximum average degree to the degree of the forest component.

Results apply to a range of M values, broadening applicability of graph partitioning techniques.

Abstract

An $({\cal I},{\cal F}_d)$-partition of a graph is a partition of the vertices of the graph into two sets $I$ and $F$, such that $I$ is an independent set and $F$ induces a forest of maximum degree at most $d$. We show that for all $M<3$ and $d \ge \frac{2}{3-M} - 2$, if a graph has maximum average degree less than $M$, then it has an $({\cal I},{\cal F}_d)$-partition. Additionally, we prove that for all $\frac{8}{3} \le M < 3$ and $d \ge \frac{1}{3-M}$, if a graph has maximum average degree less than $M$ then it has an $({\cal I},{\cal F}_d)$-partition.