Small Universal Point Sets for k-Outerplanar Graphs

Small Universal Point Sets for k-Outerplanar Graphs A point set $$\mathcal{S} \subseteq \mathbb {R}^2$$ S ⊆ R 2 is universal for a class $$\mathcal G$$ G of planar graphs if every graph of $$\mathcal{G}$$ G has a planar straight-line embedding on $$\mathcal{S}$$ S . It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an $$O(n \log n)$$ O ( n log n ) -size universal point set in two distinct cases, namely when $$k=2$$ k = 2 (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete & Computational Geometry Springer Journals
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Publisher
Springer US
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Combinatorics; Computational Mathematics and Numerical Analysis
ISSN
0179-5376
eISSN
1432-0444
D.O.I.
10.1007/s00454-018-0009-x
Publisher site
See Article on Publisher Site

Abstract

A point set $$\mathcal{S} \subseteq \mathbb {R}^2$$ S ⊆ R 2 is universal for a class $$\mathcal G$$ G of planar graphs if every graph of $$\mathcal{G}$$ G has a planar straight-line embedding on $$\mathcal{S}$$ S . It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an $$O(n \log n)$$ O ( n log n ) -size universal point set in two distinct cases, namely when $$k=2$$ k = 2 (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).

Journal

Discrete & Computational GeometrySpringer Journals

Published: May 29, 2018

References

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