eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-06-09
44:1
44:15
10.4230/LIPIcs.SoCG.2023.44
article
On the Geometric Thickness of 2-Degenerate Graphs
Jain, Rahul
1
https://orcid.org/0000-0002-8567-9475
Ricci, Marco
1
https://orcid.org/0000-0002-4502-8571
Rollin, Jonathan
1
https://orcid.org/0000-0002-6769-7098
Schulz, André
1
https://orcid.org/0000-0002-2134-4852
FernUniversität in Hagen, Germany
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].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol258-socg2023/LIPIcs.SoCG.2023.44/LIPIcs.SoCG.2023.44.pdf
Degeneracy
geometric thickness
geometric arboricity