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Note on Min- k-Planar Drawings of Graphs

Authors Petr Hliněný , Lili Ködmön

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational geometry
Keywords
  • Crossing Number
  • Planarity
  • k-Planar Graph
  • Min-k-Planar Graph

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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.

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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) https://doi.org/10.4230/LIPIcs.GD.2024.8

Author Details

Petr Hliněný
  • Masaryk University, Brno, Czech Republic
Lili Ködmön
  • Eötvös Loránd University, Budapest, Hungary

Funding

  • Ködmön, Lili: Supported by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the ELTE TKP 2021-NKTA-62 funding scheme.

References

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