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The 2-dimensional rigidity of certain families of graphs
Article first published online: 19 SEP 2006
DOI: 10.1002/jgt.20196
Copyright © 2006 Wiley Periodicals, Inc.
1097-0118/asset/cover.gif?v=1&s=747ee645df081de59c78864878c8bdb9daeca9a1 "Journal of Graph Theory")
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How to Cite
Jackson, B., Servatius, B. and Servatius, H. (2007), The 2-dimensional rigidity of certain families of graphs. J. Graph Theory, 54: 154–166. doi: 10.1002/jgt.20196
Publication History
- Issue published online: 8 DEC 2006
- Article first published online: 19 SEP 2006
- Manuscript Revised: 26 MAY 2006
- Manuscript Received: 27 JAN 2005
- Abstract
- References
- Cited By
Keywords:
- generic rigidity;
- global rigidity;
- 2-dimensional generic rigidity matroid;
- framework;
- random regular graph;
- vertex-transitive graph
Abstract
Laman's characterization of minimally rigid 2-dimensional generic frameworks gives a matroid structure on the edge set of the underlying graph, as was first pointed out and exploited by L. Lovász and Y. Yemini. Global rigidity has only recently been characterized by a combination of two results due to T. Jordán and the first named author, and R. Connelly, respectively. We use these characterizations to investigate how graph theoretic properties such as transitivity, connectivity and regularity influence (2-dimensional generic) rigidity and global rigidity and apply some of these results to reveal rigidity properties of random graphs. In particular, we characterize the globally rigid vertex transitive graphs, and show that a random d-regular graph is asymptotically almost surely globally rigid for all d ≥ 4. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 154–166, 2007