Globally rigid graphs are fully reconstructible

TL;DR

This paper proves that globally rigid graphs in or d-dimensional space are fully reconstructible from edge lengths alone, strengthening previous results and providing new families of such graphs.

Contribution

It establishes that globally rigid graphs are fully reconstructible in or rom edge lengths, removing previous vertex number constraints, and introduces new families of reconstructible graphs.

Findings

Globally rigid graphs with at least d+2 vertices are fully reconstructible in or rom edge lengths.

The d-dimensional generic rigidity matroid of such graphs is connected.

New families of fully reconstructible graphs are identified.

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.