On Graph Thickness, Geometric Thickness, and Separator Theorems

We investigate the relationship between geometric thickness and the thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O(log n). The technique used can be extended to other classes of graphs so long as a standard separator theorem exists. For example, we can apply it to show the known bound that thickness two graphs have geometric thickness O( √ n), yielding a simple construction in the process.