# Partitioning sparse graphs into an independent set and a graph with bounded size components

## TL;DR

This paper investigates how to partition graphs into an independent set and a subgraph with small components, providing bounds based on maximum average degree and girth, with implications for planar graphs.

## Contribution

It introduces new bounds for partitioning graphs into an independent set and a bounded component subgraph, including specific results for planar graphs with large girth.

## Key findings

- Graphs with mad < 2.5 admit an (I, O_3)-partition.
- Planar graphs with girth ≥ 10 can be partitioned into an independent set and paths of length at most 3.
- Graphs with mad < 8k/(3k+1) admit an (I, O_k)-partition.

## Abstract

We study the problem of partitioning the vertex set of a given graph so that each part induces a graph with components of bounded order; we are also interested in restricting these components to be paths. In particular, we say a graph $G$ admits an $({\cal I}, {\cal O}_k)$-partition if its vertex set can be partitioned into an independent set and a set that induces a graph with components of order at most $k$. We prove that every graph $G$ with $\operatorname{mad}(G)<\frac 52$ admits an $({\cal I}, {\cal O}_3)$-partition. This implies that every planar graph with girth at least $10$ can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 3. We also prove that every graph $G$ with $\operatorname{mad}(G) < \frac{8k}{3k+1} = \frac{8}{3}\left( 1 - \frac{1}{3k+1} \right)$ admits an $({\cal I}, {\cal O}_k)$-partition. This implies that every planar graph with girth at least $9$ can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 9.