The General Position Problem: A Survey

TL;DR

This survey reviews the recent developments in the general position problem in graph theory, exploring exact results, variations, and behavior under graph operations, highlighting its rapid growth and diverse applications.

Contribution

It compiles and discusses the extensive literature on the general position problem, including new variants and the effects of graph operations, serving as a comprehensive overview.

Findings

Exact results for various graph classes

Behavior of the general position number under graph operations

Discussion of multiple variations and applications

Abstract

Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for a largest set $S$ of vertices in a graph such that no three elements of $S$ lie on a common shortest path. The number of vertices in such a largest set is the \emph{general position number} of the graph. This paper provides a survey of this rapidly growing problem, which now has an extensive literature. We cover exact results for various graph classes and the behaviour of the general position number under graph products and operations. We also discuss interesting variations of the general position problem, including those corresponding to different graph convexities, as well as dynamic, fractional, colouring and game versions of the problem.