Free Sets in Planar Graphs: History and Applications
Vida Dujmović, Pat Morin
TL;DR
This survey analyzes free sets in planar graphs through four equivalent definitions (proper-good, collinear, free-collinear, free) and connects these combinatorial concepts to practical graph-drawing problems. It provides tight links between free sets and drawing tasks such as untangling, universal point subsets, and simultaneous embeddings, using tools from level-planarity, canonical orderings, and dual circumference arguments to derive large free-set bounds for broad graph classes. The work also introduces a one-bend variant of free sets, shows substantial lower and upper bounds, and highlights key open problems in growth rates, degree and treewidth restrictions, and the interplay between combinatorial and geometric perspectives. Overall, the paper offers a cohesive framework that unifies several seemingly disparate drawing problems under the theory of free sets in planar graphs and lays out rich directions for future research.
Abstract
A subset $S$ of vertices in a planar graph $G$ is a free set if, for every set $P$ of $|S|$ points in the plane, there exists a straight-line crossing-free drawing of $G$ in which vertices of $S$ are mapped to distinct points in $P$. In this survey, we review - several equivalent definitions of free sets, - results on the existence of large free sets in planar graphs and subclasses of planar graphs, - and applications of free sets in graph drawing. The survey concludes with a list of open problems in this still very active research area.