Rigidity condition for gluing two bar-joint rigid graphs embedded in $\mathbb{R}^d$

TL;DR

This paper introduces a new theorem for gluing two rigid bar-joint graphs in higher dimensions, extending combinatorial rigidity theory beyond two dimensions and enabling scalable computational algorithms.

Contribution

It presents a novel rigidity condition for combining two graphs in higher dimensions, generalizing Tay's theorem and incorporating pinned rigid graphs for overlapping vertices.

Findings

Theorem for gluing two rigid graphs in higher dimensions.

Reduction to Tay's theorem when no overlapping vertices.

Algorithmic framework for constructing rigid clusters recursively.

Abstract

How does one determine if a collection of bars joined by freely rotating hinges cannot be deformed without changing the length of any of the bars? In other words, how does one determine if a bar-joint graph is rigid? This question has been definitively answered using combinatorial rigidity theory in two dimensions via the Geiringer-Laman Theorem. However, it has not yet been answered using combinatorial rigidity theory in higher dimensions, given known counterexamples to the trivial dimensional extension of the Geiringer-Laman Theorem. To work towards a combinatorial approach in dimensions beyond two, we present a theorem for gluing two rigid bar-joint graphs together that remain rigid. When there are no overlapping vertices between the two graphs, the theorem reduces to Tay's theorem used to identify rigidity in body-bar graphs. When there are overlapping vertices, we rely on the notion of pinned rigid graphs to identify and constrain rigid motions. This theorem provides a basis for an algorithm for recursively constructing rigid clusters that can be readily adapted for computational purposes. By leveraging Henneberg-type operations to grow a rigid (or minimally rigid) graph and treating simplices-where every vertex connects to every other vertex-as fundamental units, our approach offers a scalable solution with computational complexity comparable to traditional methods. Thus, we provide a combinatorial blueprint for algorithms in multi-dimensional rigidity theory as applied to bar-joint graphs.