A Universal Point Set for 2-Outerplanar Graphs

TL;DR

This paper introduces a universal point set of size O(n log n) for 2-outerplanar graphs, advancing the understanding of minimal universal point sets for specific graph classes.

Contribution

It provides the first known universal point set of sub-quadratic size for 2-outerplanar graphs, a significant step in graph drawing research.

Findings

Universal point set size for 2-outerplanar graphs is O(n log n).

The result improves upon previous bounds for similar graph classes.

Supports efficient planar embeddings for 2-outerplanar graphs.

Abstract

A point set $S \subseteq \mathbb{R}^2$ is universal for a class $\cal G$ if every graph of ${\cal G}$ has a planar straight-line embedding on $S$. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a sub-quadratic universal point set for them is one of the most fascinating open problems in Graph Drawing. Motivated by the fact that outerplanarity is a key property for the existence of small universal point sets, we study 2-outerplanar graphs and provide for them a universal point set of size $O(n \log n)$.