Recognizing Penny and Marble Graphs is Hard for Existential Theory of the Reals

2025-08-13

AI Summary

This paper investigates the computational complexity of recognizing penny graphs (contact graphs of non-overlapping unit disks in the plane) and marble graphs (contact graphs of non-overlapping unit balls in 3D). Using constructions of systems of real algebraic constraints and establishing polynomial-time reductions to the existential theory of the reals (∃ℝ), the authors prove for the first time that both recognition problems are ∃ℝ-complete—even when the input graph is given with a fixed planar embedding. The 3D extension resolves a long-standing open problem in the complexity classification of marble graphs. Additionally, the paper shows that deciding rigidity of penny graphs is ∀ℝ-complete. By integrating tools from combinatorial geometry, graph embeddings, and real algebraic geometry, the work settles fundamental complexity questions in geometric graph theory and provides tight complexity characterizations for these central problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We show that the recognition problem for penny graphs (contact graphs of unit disks in the plane) is $existsmathbb{R}$-complete, that is, computationally as hard as the existential theory of the reals, even if a combinatorial plane embedding of the graph is given. The exact complexity of the penny graph recognition problem has been a long-standing open problem. We lift the penny graph result to three dimensions and show that the recognition problem for marble graphs (contact graphs of unit balls in three dimensions) is $existsmathbb{R}$-complete. Finally, we show that rigidity of penny graphs is $forallmathbb{R}$-complete and look at grid embeddings of penny graphs that are trees.

Problem

• Penny graph recognition is ∃ℝ-complete
• Marble graph recognition is ∃ℝ-complete
• Penny graph rigidity is ∀ℝ-complete

Innovation

• Proves penny graph recognition is ∃ℝ-complete
• Extends result to 3D marble graphs
• Shows penny graph rigidity is ∀ℝ-complete