Generic Rigidity of Graph Frameworks in Euclidean Space
Abstract
The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer and independently in 1970 by Laman, analogous edge-density counts on subgraphs fail in higher dimensions. In this paper, we solve the problem by providing a combinatorial characterization of generic infinitesimal rigidity valid in all dimensions. By gluing together local versions of Cramer's rule at each vertex, we construct a globally valid self-stress on the edges. The compatibility conditions governing these local solutions are controlled by the Pl\"ucker relations on the Grassmannian , allowing us to check generic rigidity using the combinatorics of Young's straightening law on tableaux.
Keywords
Cite
@article{arxiv.2604.05442,
title = {Generic Rigidity of Graph Frameworks in Euclidean Space},
author = {Alexander Heaton},
journal= {arXiv preprint arXiv:2604.05442},
year = {2026}
}
Comments
Simplified results using multi-homogeneity. Expanded introduction and references. Added context on Plucker relations, showing how to straighten the running example by hand. Applied LGV lemma to reformulate T_\sigma