Planar graphs without $5^{-}$-cycles at distance less than $3$ are $(\mathcal{I},\mathcal{F})$-colorable

A graph is $(\mathcal{I},\mathcal{F})$-colorable if its vertex set can be partitioned into two subsets, one of which is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without $5^{-}$-cycles at distance less than $3$ is $(\mathcal{I},\mathcal{F})$-colorable.