Local and global $d$-rigidity are not definable in the first order logic of graphs
Daniel Irving Bernstein, Nathaniel Vaduthala
TL;DR
The paper investigates whether local and global $d$-rigidity can be defined in the first-order theory of graphs. It combines Hanf locality with a rigidity result for $k$-circuits to show non-definability, constructing graphs with identical $r$-neighborhoods but differing rigidity properties. Central to the argument is the use of $\mathcal{C}_n^d$ and $\mathcal{P}_n^d$ simplicial complexes and the Cruickshank–Jackson–Tanigawa theorem on global rigidity of triangulated manifolds. The result generalizes the classical fact that graph connectivity is not FO-definable and demonstrates fundamental limits on describing rigidity combinatorially in FO logic.
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.