  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.

Discrete Mathematics;Combinatorics