Globally rigid graphs are fully reconstructible

Abstract

AbstractAd-dimensional framework is a pair$(G,p)$, where$G=(V,E)$is a graph andpis a map fromVto$\mathbb {R}^d$. 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 graphGand its edge lengths uniquely determine$(G,p)$, up to congruence. A graphGis called globally rigid in$\mathbb {R}^d$if everyd-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 graphGis strongly reconstructible in$\mathbb {C}^d$if the set of (unlabeled) edge lengths of any generic framework$(G,p)$ind-space, along with the number of vertices ofG, uniquely determine bothGand the association between the edges ofGand the set of edge lengths. It is known that ifGis 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,Gis in fact fully reconstructible in$\mathbb {C}^d$, which means that the set of edge lengths alone is sufficient to uniquely reconstructG, 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 ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then thed-dimensional generic rigidity matroid ofGis connected. Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.