Minimally globally rigid graphs

TL;DR

The paper investigates minimally globally rigid graphs in Euclidean space and proves a tight edge bound: for a graph $G$ minimally globally rigid in $\mathbb{R}^d$ with at least $d+2$ vertices, $|E|\le (d+1)|V| - \binom{d+2}{2}$, with the minimum degree bounded by $2d+1$ and equality only for $K_{d+2}$. It further shows that such graphs are flexible in $\mathbb{R}^{d+1}$ when $|V|\ge d+3$, and that dense graphs in $\mathbb{R}^2$ necessarily contain nontrivial globally rigid subgraphs. The work leverages the rigidity matroid, stress matrices, and a rank-sum lemma to connect $d$-dimensional global rigidity with $(d+1)$-dimensional rigidity, and explores conjectures about globally linked pairs and higher-dimensional generalizations via coning and mixed connectivity. Overall, the results clarify the sparsity structure of minimally globally rigid graphs, reveal a dimension-dropping relationship between $d$- and $(d+1)$-dimensional rigidity, and provide a framework for identifying globally rigid subgraphs in dense graphs, with several conjectures guiding future work.

Abstract

A graph $G = (V,E)$ is globally rigid in $\mathbb{R}^d$ if for any generic placement $p : V \rightarrow \mathbb{R}^d$ of the vertices, the edge lengths $||p(u) - p(v)||, uv \in E$ uniquely determine $p$, up to congruence. In this paper we consider minimally globally rigid graphs, in which the deletion of an arbitrary edge destroys global rigidity. We prove that if $G=(V,E)$ is minimally globally rigid in $\mathbb{R}^d$ on at least $d+2$ vertices, then $|E|\leq (d+1)|V|-\binom{d+2}{2}$. This implies that the minimum degree of $G$ is at most $2d+1$. We also show that the only graph in which the upper bound on the number of edges is attained is the complete graph $K_{d+2}$. It follows that every minimally globally rigid graph in $\mathbb{R}^d$ on at least $d+3$ vertices is flexible in $\mathbb{R}^{d+1}$. As a counterpart to our main result on the sparsity of minimally globally rigid graphs, we show that in two dimensions, dense graphs always contain nontrivial globally rigid subgraphs. More precisely, if some graph $G=(V,E)$ satisfies $|E|\geq 5|V|$, then $G$ contains a subgraph on at least seven vertices that is globally rigid in $\mathbb{R}^2$. If the well-known "sufficient connectivity conjecture" is true, then our methods also extend to higher dimensions. Finally, we discuss a conjectured strengthening of our main result, which states that if a pair of vertices $\{u,v\}$ is linked in $G$ in $\mathbb{R}^{d+1}$, then $\{u,v\}$ is globally linked in $G$ in $\mathbb{R}^d$. We prove this conjecture in the $d=1,2$ cases, along with a variety of related results.

Figures (4)

• Figure 1: The skeleton of the icosahedron with an additional "bracing" edge. This graph is minimally globally rigid in $\mathbb{R}^3$ and has minimum degree $5$.
• Figure 2: The $2$-sum operation on two copies of $K_4$.
• Figure 3: A pair of isomorphic graphs that are not redundantly $\mathcal{R}_1$-connected, but their $2$-sum is redundantly $\mathcal{R}_1$-connected.
• Figure 4: Gluing $\ell$ copies of $K_4 - e$ ($\ell=3$ in the drawing) gives a graph with no nontrivial globally rigid subgraphs in $\mathbb{R}^2$, on $2\ell + 2$ vertices and with $5\ell$ edges.

Theorems & Definitions (67)

• Conjecture 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9