The slow-coloring game on sparse graphs: $k$-degenerate, planar, and outerplanar

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, marks nonempty set $M$ uncolored vertices, colors subset that independent in game ends when colored. score the sum sizes sets marked Lister. goal to minimize score, while tries maximize it. We provide strategies for various classes graphs whose can be partitioned into bounded number inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, outerplanar graphs. For example, we show an $n$-vertex $G$, keep at most $\frac{3k+4}4n$ $3.9857n$ $5$-colorable, $3n$ planar with Hamiltonian dual, $\frac{8n+3m}5$ $4$-colorable $m$ edges (hence $3.4n$ planar), $\frac73n$ outerplanar.