How many vertex locations can be arbitrarily chosen when drawing planar graphs?

TL;DR

This paper establishes bounds on the number of arbitrary vertex locations in planar graph drawings, providing constructive proofs and efficient algorithms for embedding such points.

Contribution

It introduces new bounds on the size of point sets that can be vertices of planar graphs and provides constructive, linear-time drawing algorithms.

Findings

Sets of size roughly √(log n) can be vertex sets in any planar graph drawing.

One-sided convex point sets can host about n^(1/3) vertices.

Constructive proofs lead to O(n)-time drawing algorithms.

Abstract

It is proven that every set $S$ of distinct points in the plane with cardinality $\lceil \frac{\sqrt{\log_2 n}-1}{4} \rceil$ can be a subset of the vertices of a crossing-free straight-line drawing of any planar graph with $n$ vertices. It is also proven that if $S$ is restricted to be a one-sided convex point set, its cardinality increases to $\lceil \sqrt[3]{n} \rceil$. The proofs are constructive and give rise to O(n)-time drawing algorithms. As a part of our proofs, we show that every maximal planar graph contains a large induced biconnected outerplanar graphs and a large induced outerpath (an outerplanar graph whose weak dual is a path).