A d-dimensional framework is a pair (G, p), where $$G=(V,E)$$ G = ( V , E ) is a graph and p is a map from V to $$\mathbb {R}^d$$ R d . The length of an edge $$uv\in E$$ u v ∈ E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in $$\mathbb {R}^d$$ R d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in $$\mathbb {R}^d$$ R d is globally rigid for some graph G on $$n\ge d+2$$ n ≥ d + 2 vertices (where $$d\ge 2$$ d ≥ 2 ), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For $$d=1,2$$ d = 1 , 2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid.

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.

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