Globally rigid graphs are fully reconstructible

TL;DR

This work studies the problem of reconstructing a graph from edge lengths of a generic $d$-dimensional framework by introducing and analyzing the $d$-dimensional measurement variety $M_{d,G}$. It establishes that if a graph $G$ is globally rigid in $\,\mathbb{R}^d$ on at least $d+2$ vertices, then $G$ is fully reconstructible in $\,\mathbb{C}^d$ (i.e., edge lengths alone determine $G$ up to isomorphism, without a fixed vertex count). A key technical advance is proving that globally rigid graphs are $\,\mathcal{R}_d$-connected, which underpins the full reconstructibility result and enables the construction of new fully reconstructible graphs. The authors also develop a rich algebraic-geometry framework around measurement varieties, Gauss fibers, and coning to relate combinatorial properties with algebraic ones, and they address questions on unlabeled reconstructibility and graph composition, showing both positive examples and limits. Overall, the paper advances our understanding of when edge-length data suffice to uniquely determine a graph in higher dimensions, with implications for rigidity theory and graph reconstruction problems.

Abstract

A $d$-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and $p$ is a map from $V$ to $\mathbb{R}^d$. The length of an edge $uv\in E$ in $(G,p)$ is the distance between $p(u)$ and $p(v)$. The framework is said to be globally rigid in $\mathbb{R}^d$ if the graph $G$ and its edge lengths uniquely determine $(G,p)$, up to congruence. A graph $G$ is called globally rigid in $\mathbb{R}^d$ if every $d$-dimensional generic framework $(G,p)$ is globally rigid. In this paper, we consider the problem of reconstructing a graph from the set of edge lengths arising from a generic framework. Roughly speaking, a graph $G$ is strongly reconstructible in $\mathbb{C}^d$ if the set of (unlabeled) edge lengths of any generic framework $(G,p)$ in $d$-space, along with the number of vertices of $G$, uniquely determine both $G$ and the association between the edges of $G$ and the set of edge lengths. It is known that if $G$ is globally rigid in $\mathbb{R}^d$ on at least $d+2$ vertices, then it is strongly reconstructible in $\mathbb{C}^d$. We strengthen this result and show that under the same conditions, $G$ is in fact fully reconstructible in $\mathbb{C}^d$, which means that the set of edge lengths alone is sufficient to uniquely reconstruct $G$, without any constraint on the number of vertices (although still under the assumption that the edge lengths come from a generic realization). As a key step in our proof, we also prove that if $G$ is globally rigid in $\mathbb{R}^d$ on at least $d+2$ vertices, then the $d$-dimensional generic rigidity matroid of $G$ is connected. Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.

Figures (3)

• Figure 1: A $3$-connected, redundantly rigid and $M$-separable graph in $\mathbb{R}^3$. This graph satisfies $r_3(G)= 36 = 27 + 9 =r_3(G^o)+r_3(K_5)$, where $G^o$ is the outer ring of $K_5$'s and $K_5$ is the subgraph induced by the black (filled) vertices.
• Figure 2: Graphs that are $4$-connected, redundantly rigid and $M$-separable in $\mathbb{R}^3$. The graph shown in (a) satisfies $r_3(G)= 105 = 96 + 9 =r_3(G^o)+r_3(K_5)$, where $G^o$ is the outer ring of $K_5$'s.
• Figure 3: A graph that is $4$-connected, redundantly rigid in $\mathbb{R}^3$ and $M$-connected in $\mathbb{R}^3$, but not globally rigid in $\mathbb{R}^3$.

Theorems & Definitions (74)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• Lemma 2.7
• proof
• Theorem 2.8