On the geometric thickness of 2-degenerate graphs
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].
Keywords
Cite
@article{arxiv.2302.14721,
title = {On the geometric thickness of 2-degenerate graphs},
author = {Rahul Jain and Marco Ricci and Jonathan Rollin and André Schulz},
journal= {arXiv preprint arXiv:2302.14721},
year = {2023}
}
Comments
25 pages, 23 figures. Full version of extended abstracts appearing at EuroCG 2023 and SoCG 2023