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Note on Min- k-Planar Drawings of Graphs

Authors: Petr Hliněný and Lili Ködmön

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)


Abstract
The k-planar graphs, which are (usually with small values of k such as 1,2,3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple. In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k ≤ 3 known (but perhaps not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k ≤ 4, the difference between general and simple drawings in the latter concept is more striking. We prove that there exist graphs with a min-2-planar drawing, or with a min-3-planar drawing avoiding crossings of adjacent edges, which have no simple min-k-planar drawings for arbitrarily large fixed k.

Cite as

Petr Hliněný and Lili Ködmön. Note on Min- k-Planar Drawings of Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 8:1-8:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{hlineny_et_al:LIPIcs.GD.2024.8,
  author =	{Hlin\v{e}n\'{y}, Petr and K\"{o}dm\"{o}n, Lili},
  title =	{{Note on Min- k-Planar Drawings of Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{8:1--8:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.8},
  URN =		{urn:nbn:de:0030-drops-212924},
  doi =		{10.4230/LIPIcs.GD.2024.8},
  annote =	{Keywords: Crossing Number, Planarity, k-Planar Graph, Min-k-Planar Graph}
}
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