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DOI: 10.4230/LIPIcs.SoCG.2023.44

On the Geometric Thickness of 2-Degenerate Graphs

Authors: Rahul Jain, Marco Ricci, Jonathan Rollin, and André Schulz

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

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].

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Rahul Jain, Marco Ricci, Jonathan Rollin, and André Schulz. On the Geometric Thickness of 2-Degenerate Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jain_et_al:LIPIcs.SoCG.2023.44,
  author =	{Jain, Rahul and Ricci, Marco and Rollin, Jonathan and Schulz, Andr\'{e}},
  title =	{{On the Geometric Thickness of 2-Degenerate Graphs}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{44:1--44:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.44},
  URN =		{urn:nbn:de:0030-drops-178946},
  doi =		{10.4230/LIPIcs.SoCG.2023.44},
  annote =	{Keywords: Degeneracy, geometric thickness, geometric arboricity}
}

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On the Geometric Thickness of 2-Degenerate Graphs

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