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

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 $(\mathcal{I},{\mathcal{O}}_k)$-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 $\mathrm{mad}(G)<\frac{5}{2}$ admits an $(\mathcal{I},{\mathcal{O}}_3)$-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 $\operatorname{mad}(G) < \frac{8k}{3k+1} = \frac{8}{3}\left( 1 - \frac{1}{3k+1} \right)$admitsan$({\cal I}, {\cal O}_k)$-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.