Free Sets in Planar Graphs: History and Applications

2024-03-25

arXiv.org

AI Summary

This paper investigates the free set problem for planar graphs: a vertex subset $ S $ is a *free set* if, for any set of $ |S| $ points in the plane, there exists a crossing-free straight-line embedding of the graph mapping $ S $ bijectively onto those points. We unify multiple equivalent characterizations of free sets, integrating combinatorial graph theory, computational geometry, and planar embedding theory. Our analysis establishes quantitative relationships between the size of maximum free sets and structural graph parameters—including maximum degree, connectivity, and outerplanarity. We construct tight lower bounds on the size of optimal free sets for several graph classes, notably outerplanar graphs and 2-trees. Furthermore, we identify novel applications of free sets in dynamic graph drawing and geometric graph embeddings. Finally, we compile over ten key open problems to guide systematic advancement of the theory. (149 words)

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Define free sets in planar graphs mathematically
• Explore existence of large free sets in planar graphs
• Apply free sets to graph drawing techniques

Innovation

• Defines free sets in planar graphs
• Explores large free sets existence
• Applies free sets in graph drawing