Partitioning planar graphs with girth at least $9$ into an edgeless graph and a graph with bounded size components

In this paper, we study the problem of partitioning the vertex set of a planar graph with girth restriction into parts, also referred to as color classes, such that each part induces a graph with components of bounded order. An ( $\mathcal{I}$, ${\mathcal{O}}_k$)-partition of a graph $G$ is the partition of $V(G)$ into two non-empty subsets $V_1$ and $V_2$, such that $G[V_1]$ is an edgeless graph and $G[V_2]$ is a graph with components of order at most $k$. We prove that every planar graph with girth 9 and without intersecting $9$-face admits an ( $\mathcal{I}$, ${\mathcal{O}}_6$)-partition. This improves a result of Choi, Dross and Ochem (2020) which says every planar graph with girth at least $9$ admits an ( $\mathcal{I}$, ${\mathcal{O}}_9$)-partition.