On the geometric thickness of 2-degenerate graphs

A graph is $2$-degenerate if every subgraph contains a vertex of degree at most $2$. We show that every $2$-degenerate graph can be drawn with straight lines such that the drawing decomposes into $4$ plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of $2$-degenerate graphs is at most $4$. On the other hand, we show that there are $2$-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into $2$ plane graphs. That is, there are $2$-degenerate graphs with geometric thickness, and hence geometric arboricity, at least $3$. This answers two questions posed by Eppstein [Separating thickness from geometric thickness. In: Towards a Theory of Geometric Graphs, volume 342 of Contemp. Math., AMS, 2004].