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# Linear Size Universal Point Sets for Classes of Planar Graphs

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LIPIcs.SoCG.2023.31.pdf
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## Acknowledgements

We are highly indebted to Henry Förster, Linda Kleist, Joachim Orthaber and Marco Ricci due to discussions during GG-Week 2022 resulting in a solution to the problem of separating 2-cycles in our proof for subcubic graphs.

## Cite As

Stefan Felsner, Hendrik Schrezenmaier, Felix Schröder, and Raphael Steiner. Linear Size Universal Point Sets for Classes of Planar Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.31

## Abstract

A finite set P of points in the plane is n-universal with respect to a class 𝒞 of planar graphs if every n-vertex graph in 𝒞 admits a crossing-free straight-line drawing with vertices at points of P. For the class of all planar graphs the best known upper bound on the size of a universal point set is quadratic and the best known lower bound is linear in n. Some classes of planar graphs are known to admit universal point sets of near linear size, however, there are no truly linear bounds for interesting classes beyond outerplanar graphs. In this paper, we show that there is a universal point set of size 2n-2 for the class of bipartite planar graphs with n vertices. The same point set is also universal for the class of n-vertex planar graphs of maximum degree 3. The point set used for the results is what we call an exploding double chain, and we prove that this point set allows planar straight-line embeddings of many more planar graphs, namely of all subgraphs of planar graphs admitting a one-sided Hamiltonian cycle. The result for bipartite graphs also implies that every n-vertex plane graph has a 1-bend drawing all whose bends and vertices are contained in a specific point set of size 4n-6, this improves a bound of 6n-10 for the same problem by Löffler and Tóth.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
• Mathematics of computing → Graphs and surfaces
• Human-centered computing → Graph drawings
• Theory of computation → Computational geometry
• Theory of computation → Randomness, geometry and discrete structures
##### Keywords
• Graph drawing
• Universal point set
• One-sided Hamiltonian
• 2-page book embedding
• Separating decomposition
• 2-tree
• Subcubic planar graph

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