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The Number of Edges in Maximal 2-Planar Graphs

Authors Michael Hoffmann , Meghana M. Reddy

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Author Details

Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Meghana M. Reddy
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

Supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681.

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Michael Hoffmann and Meghana M. Reddy. The Number of Edges in Maximal 2-Planar Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.39

Abstract

A graph is 2-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal 2-planar if no edge can be added such that the resulting graph remains 2-planar. A 2-planar graph on n vertices has at most 5n-10 edges, and some (maximal) 2-planar graphs - referred to as optimal 2-planar - achieve this bound. However, in strong contrast to maximal planar graphs, a maximal 2-planar graph may have fewer than the maximum possible number of edges. In this paper, we determine the minimum edge density of maximal 2-planar graphs by proving that every maximal 2-planar graph on n ≥ 5 vertices has at least 2n edges. We also show that this bound is tight, up to an additive constant. The lower bound is based on an analysis of the degree distribution in specific classes of drawings of the graph. The upper bound construction is verified by carefully exploring the space of admissible drawings using computer support.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Human-centered computing → Graph drawings
Keywords
  • k-planar graphs
  • local crossing number
  • saturated graphs
  • beyond-planar graphs

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References

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