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Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

The Rigidity Transition in Random Graphs

Abstract

As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdős-Rényi random graph G(n, c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c2, w.h.p. all rigid components span one, two, or three vertices, and when c > c2, w.h.p. there is a giant rigid component. The constant c2 ≈ 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA'07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1 − o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.

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Published In

cover image Proceedings
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 1237 - 1252
Editor: Dana Randall, Georgia Institute of Technology, Atlanta, Georgia
ISBN (Print): 978-0-898719-93-2
ISBN (Online): 978-1-61197-308-2

History

Published online: 18 December 2013

Authors

Affiliations

Shiva Prasad Kasiviswanathan*

Notes

*
Work done while the author was as a postdoc at Los Alamos National Laboratory
Supported by CDI-I grant DMR 0835586 to I. Rivin and M. M. J. Treacy.

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