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The General Position Problem: A Survey

Ullas Chandran S. V., Sandi Klavžar, James Tuite

TL;DR

The survey consolidates the rich, rapidly expanding literature on the general position problem in graphs, tracing its origins to classical no-three-in-line puzzles and advancing through precise bounds, complexity results, and exact values for diverse graph classes and products. It organizes distance-based and non-distance variants, and introduces numerous generalizations (monophonic, Steiner, detour) as well as dynamic and fractional versions, emphasizing structural characterizations via isometric covers and strong resolving graphs. A core contribution is mapping extensive product behavior (Cartesian, strong, direct, Sierpiński, lexicographic) and detailing how gp-values interact with graph operations, while highlighting open problems and computational complexities (notably NP-hardness and PSPACE-completeness) that guide future work. The work also connects gp-numbers to broader graph-convexity concepts, graph polynomials, and combinatorial optimization, underlining practical applications in robotics, network visibility, and algorithmic graph theory.

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.

The General Position Problem: A Survey

TL;DR

The survey consolidates the rich, rapidly expanding literature on the general position problem in graphs, tracing its origins to classical no-three-in-line puzzles and advancing through precise bounds, complexity results, and exact values for diverse graph classes and products. It organizes distance-based and non-distance variants, and introduces numerous generalizations (monophonic, Steiner, detour) as well as dynamic and fractional versions, emphasizing structural characterizations via isometric covers and strong resolving graphs. A core contribution is mapping extensive product behavior (Cartesian, strong, direct, Sierpiński, lexicographic) and detailing how gp-values interact with graph operations, while highlighting open problems and computational complexities (notably NP-hardness and PSPACE-completeness) that guide future work. The work also connects gp-numbers to broader graph-convexity concepts, graph polynomials, and combinatorial optimization, underlining practical applications in robotics, network visibility, and algorithmic graph theory.

Abstract

Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for a largest set of vertices in a graph such that no three elements of 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.

Paper Structure

This paper contains 42 sections, 121 theorems, 77 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Terminology
  3. General Position Number
  4. Bounds
  5. Strong resolving graphs
  6. Complexity
  7. Exact results
  8. Kneser graphs
  9. Maximal outerplanar graphs
  10. Additional exact results
  11. General position polynomials
  12. Graph products
  13. Cartesian product
  14. Strong product and direct product
  15. Generalised lexicographic product
  16. ...and 27 more sections

Key Result

Theorem 2.1

AnaChaChaKlaTho Let $G$ be a connected graph. Then $X\subseteq V(G)$ is a general position set if and only if the components of $G[X]$ are cliques, the vertices of which form an in-transitive, distance-constant partition of $X$.

Figures (8)

  • Figure 1: A no-three-in-line configuration of 120 pawns on a $60 \times 60$ chessboard from Prellberg, reproduced with kind permission from T. Prellberg (with lines connecting the vertices on the right to show the complexity of the problem)
  • Figure 2: A gp-set in the Clebsch graph
  • Figure 3: A gp-set in the Frucht graph
  • Figure 4: The Petersen graph with different types of position set. On the left in red: a largest general position set. In the centre in green: a largest monophonic position set. On the right in blue: a largest equidistant set, independent position set and mobile general position set, also a lower general position set.
  • Figure 5: A graph with order ten, gp-number three and largest size
  • ...and 3 more figures

Theorems & Definitions (133)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 123 more