Abstract
We examine the generic local and global rigidity of various graphs in ℝd . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for d>3, and we define a construction that generates infinitely many graphs in ℝ5. Finally, we state some conjectures for further exploration.
Highlights
Introduction and PreliminariesA framework consists of a graph whose vertices have been assigned coordinates in Rd
An important question is whether or not a given framework is locally rigid, that is, whether there is a way to continuously deform the framework while maintaining its edge lengths
Because Υ is a bipartite graph, there are no edges between any two vertices in Ai ∪ Ai+2 ∪ · · · ∪ Aj−1; the same can be said of the vertices in Ai+1 ∪ Ai+3 ∪ · · · ∪ Aj
Summary
A framework consists of a graph whose vertices have been assigned coordinates in Rd. An important question is whether or not a given framework is locally rigid, that is, whether there is a way to continuously deform the framework while maintaining its edge lengths. Theorem 3 (Connelly [4], Gortler–Healy–Thurston [7]) A graph with at least d + 2 vertices is generically globally rigid if and only if, for some generic realization, there is a stress matrix with nullity d + 1. Showed that this condition is sufficient; Gortler, Healy and Thurston showed that it is necessary as well, implying that global rigidity is a generic property of the graph. Theorem 6 (Connelly and Whiteley [6]) For any graph G, coning preserves the generic local, redundant, and global rigidity of G from Rd to Rd+1.
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