# On the geometric thickness of 2-degenerate graphs

## TL;DR

This paper investigates the geometric thickness of 2-degenerate graphs, establishing upper and lower bounds, and answering open questions about their decomposability into plane graphs.

## Contribution

It proves that 2-degenerate graphs have geometric thickness at most 4 and provides examples with thickness at least 3, resolving previously open questions.

## Key findings

- Every 2-degenerate graph can be drawn with geometric thickness at most 4.
- There exist 2-degenerate graphs with geometric thickness at least 3.
- The results answer open questions by Eppstein on geometric thickness bounds.

## Abstract

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, vol. 342 of Contemp. Math., AMS, 2004].