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

TL;DR

This paper proves that recognizing penny and marble graphs, contact graphs of unit disks and balls respectively, is computationally as hard as the existential theory of the reals, resolving a long-standing open problem.

Contribution

It establishes the $orall ext{R}$-completeness of penny graph rigidity and extends the $ ext{ER}$-completeness result to three-dimensional marble graphs.

Findings

Recognition of penny graphs is $ ext{ER}$-complete.

Recognition of marble graphs in 3D is $ ext{ER}$-complete.

Rigidity of penny graphs is $orall ext{R}$-complete.

Abstract

We show that the recognition problem for penny graphs (contact graphs of unit disks in the plane) is $\exists\mathbb{R}$-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 $\exists\mathbb{R}$-complete. Finally, we show that rigidity of penny graphs is $\forall\mathbb{R}$-complete and look at grid embeddings of penny graphs that are trees.