Globally Linked Pairs of Vertices in Generic Frameworks

• Published in: Combinatorica (Budapest. 1981) Vol. 44; no. 4; pp. 817 - 838
• Main Authors: Jordán, Tibor, Villányi, Soma
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2024; Springer Nature B.V
• Subjects: Algorithms; Apexes; Combinatorics; Graph theory; Graphs; Mathematics; Mathematics and Statistics; Original Paper; Globally rigid graph; Rigid graph; Globally linked pair; Framework
• ISSN: 0209-9683; 1439-6912; 1439-6912
• DOI: 10.1007/s00493-024-00094-3

A d -dimensional framework is a pair ( G ,  p ), where G = ( V , E ) is a graph and p is a map from V to R d . The length of an edge x y ∈ E in ( G ,  p ) is the distance between p ( x ) and p ( y ). A vertex pair { u , v } of G is said to be globally linked in ( G ,  p ) if the distance between p ( u ) and p ( v ) is equal to the distance between q ( u ) and q ( v ) for every d -dimensional framework ( G ,  q ) in which the corresponding edge lengths are the same as in ( G ,  p ). We call ( G ,  p ) globally rigid in R d when each vertex pair of G is globally linked in ( G ,  p ). A pair { u , v } of vertices of G is said to be weakly globally linked in G in R d if there exists a generic framework ( G ,  p ) in which { u , v } is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a ( d + 1 ) -connected graph G in R d and then show that for d = 2 it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in R 2 , which gives rise to an algorithm for testing weak global linkedness in the plane in O ( | V | 2 ) time. Our methods lead to a new short proof for the characterization of globally rigid graphs in R 2 , and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.