Local and global $d$-rigidity are not definable in the first order logic of graphs

TL;DR

This paper proves that the properties of local and global $d$-rigidity in graphs cannot be expressed using first-order logic, highlighting limitations in logical definability of these geometric graph properties.

Contribution

It establishes the non-definability of local and global $d$-rigidity in first-order graph logic using Hanf locality and existing rigidity results.

Findings

Local and global $d$-rigidity are not first-order definable.

Hanf locality and rigidity results are key tools in the proof.

The result clarifies logical limits in graph rigidity theory.

Abstract

We use Hanf locality and a result of Cruickshank, Jackson, and Tanigawa on the global rigidity of graphs of $k$-circuits to prove that local and global $d$-rigidity are not definable in the first order logic of graphs.