Globally Linked Pairs of Vertices in Rigid Frameworks

  • Bill Jackson
  • Tibor JordánEmail author
  • Zoltán Szabadka
Part of the Fields Institute Communications book series (FIC, volume 70)


A 2-dimensional framework (G, p) is a graph G = (V, E) together with a map \(p: V \rightarrow \mathbb{R}^{2}\). We consider the framework to be a straight line realization of G in \(\mathbb{R}^{2}\). Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u, v} is globally linked in G if the distance between the points corresponding to u and v is the same in all pairs of equivalent generic realizations of G.In this paper we extend our previous results on globally linked pairs and complete the characterization of globally linked pairs in minimally rigid graphs. We also show that the Henneberg 1-extension operation on a non-redundant edge preserves the property of being not globally linked, which can be used to identify globally linked pairs in broader families of graphs. We prove that if (G, p) is generic then the set of globally linked pairs does not change if we perturb the coordinates slightly. Finally, we investigate when we can choose a non-redundant edge e of G and then continuously deform a generic realization of Ge to obtain equivalent generic realizations of G in which the distances between a given pair of vertices are different.


Global rigidity Rigid framework Bar-and-joint framework Globally linked pair Minimally rigid graph 

Subject Classifications

52C25 05C10 



Section 3 is based on an earlier manuscript of one of the authors [25]. We thank Viet-Hang Nguyen for pointing out an error in [25] and for several useful discussions. We are grateful to the anonymous referee for several very useful comments. We also thank Balázs Csikós and András Szűcs for interesting discussions on the differential topology aspects of our problems and Oleg Karpenkov for suggestions which led to the example given in Fig. 4.

This work was supported by the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, the Hungarian Scientific Research Fund grant no. K81472, and the National Development Agency of Hungary, grant no. CK 80124, based on a source from the Research and Technology Innovation Fund.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueen Mary University of LondonLondonUK
  2. 2.Department of Operations ResearchEotvos University, and the MTA-ELTE Egervary Research Group on Combinatorial optimizationBudapestHungary
  3. 3.Department of Operations ResearchEötvös UniversityBudapestHungary

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