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DOI: 10.4230/LIPIcs.GD.2025.15

Crossing Number of Simple 3-Plane Drawings

Authors: Miriam Goetze, Michael Hoffmann, Ignaz Rutter, and Torsten Ueckerdt

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

We study 3-plane drawings, that is, drawings of graphs in which every edge has at most three crossings. We show how the recently developed Density Formula for topological drawings of graphs [Kaufmann et al., 2024] can be used to count the crossings in terms of the number n of vertices. As a main result, we show that every 3-plane drawing has at most 5.5(n-2) crossings, which is tight. In particular, it follows that every 3-planar graph on n vertices has crossing number at most 5.5n, which improves upon a recent bound [Bekos et al., 2024] of 6.6n. To apply the Density Formula, we carefully analyze the interplay between certain configurations of cells in a 3-plane drawing. As a by-product, we also obtain an alternative proof for the known statement that every 3-planar graph has at most 5.5(n-2) edges.

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Miriam Goetze, Michael Hoffmann, Ignaz Rutter, and Torsten Ueckerdt. Crossing Number of Simple 3-Plane Drawings. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{goetze_et_al:LIPIcs.GD.2025.15,
  author =	{Goetze, Miriam and Hoffmann, Michael and Rutter, Ignaz and Ueckerdt, Torsten},
  title =	{{Crossing Number of Simple 3-Plane Drawings}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.15},
  URN =		{urn:nbn:de:0030-drops-250014},
  doi =		{10.4230/LIPIcs.GD.2025.15},
  annote =	{Keywords: beyond planar graphs, edge density, crossing number, density formula}
}

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DOI: 10.4230/LIPIcs.ESA.2022.62

Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs

Authors: Miriam Goetze, Paul Jungeblut, and Torsten Ueckerdt

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract

It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an n-vertex planar graph, augments this graph in 𝒪(n²) steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized (Anti)factor-problem for the fixed embedding case, and SPQR-trees for the variable embedding case.

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Miriam Goetze, Paul Jungeblut, and Torsten Ueckerdt. Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 62:1-62:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goetze_et_al:LIPIcs.ESA.2022.62,
  author =	{Goetze, Miriam and Jungeblut, Paul and Ueckerdt, Torsten},
  title =	{{Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{62:1--62:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.62},
  URN =		{urn:nbn:de:0030-drops-170007},
  doi =		{10.4230/LIPIcs.ESA.2022.62},
  annote =	{Keywords: edge colorings, planar graphs, cubic graphs, generalized factors, SPQR-tree}
}

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Documents authored by Goetze, Miriam

Crossing Number of Simple 3-Plane Drawings

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Efficient Recognition of Subgraphs of Planar Cubic Bridgeless Graphs

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