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 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 \(\mathbb {R}^d\) is globally rigid for some graph G on \(n\ge d+2\) vertices (where \(d\ge 2\)), 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 \(d=1,2\) 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.

甲d维框架是一对（g ^，  p），其中\（G =（V，E）\）是曲线图和p是从地图V到 \（\ mathbb {R} ^ d \） 。（G，  p）中的边\（uv \ in E \）的长度是p（u）与p（v）之间的距离。该框架被说成是在全球范围内刚性 \（\ mathbb {R} ^ d \）如果每一个其他d维框架（g ^，  q相应的边长相同的）等于（G，  p）。在最近的论文Gortler，Theran和瑟斯顿证明，如果每一个通用框架（g ^，  p）在 \（\ mathbb {R} ^ d \）为一些图形全局刚性ģ上\（N \ GE d + 2 \ ）顶点（其中\（d \ ge 2 \）），然后已经是通用框架（G，  p）的一组（未标记的）边长，以及 n，确定框架的一致性。在本文中，当上述通用全局刚度属性不适用于该图时，我们将研究相应的未标记重构问题。我们提供了图G族，图族中d空间中任何通用框架（G，  p）的（未标记）边长的集合以及顶点的数目唯一地确定了图，直至同构。我们称这些图为弱可重构的。我们还介绍了强可重构性的概念；在这种情况下，边缘的标签也由任何通用框架的边缘长度集合确定。对于\（d = 1,2 \）我们给出了弱可重构性的部分特征以及图的强可重构性的完整特征。特别是在低维情况下，我们描述了刚性但不是冗余刚性的弱可重构图族。