Abstract
A graph \(G=(V,E)\) is called \(k\)-rigid in \(\mathbb {R}^{d}\) if \(|V|\ge k+1\) and after deleting any set of at most \(k-1\) vertices the resulting graph is rigid in \(\mathbb {R}^{d}\). A \(k\)-rigid graph \(G\) is called minimally \(k\)-rigid if the omission of an arbitrary edge results in a graph that is not \(k\)-rigid. B. Servatius showed that a 2-rigid graph in \(\mathbb {R}^2\) has at least \(2|V|-1\) edges and this bound is sharp. We extend this lower bound for arbitrary values of \(k\) and \(d\) and show its sharpness for the cases where \(k=2\) and \(d\) is arbitrary and where \(k=d=3\). We also provide a sharp upper bound for the number of edges of minimally \(k\)-rigid graphs in \(\mathbb {R}^d\) for all \(k\).
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Acknowledgments
The authors received a Grant (No. CK 80124) from the National Development Agency of Hungary, based on a source from the Research and Technology Innovation Fund. Research was supported by the MTA-ELTE Egerváry Research Group and by the Hungarian Scientific Research Fund—OTKA, K109240. The authors thank Zsuzsanna Jankó and János Geleji for the inspiring discussions and Tibor Jordán for posing the interesting questions solved in this paper.
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Kaszanitzky, V.E., Király, C. On Minimally Highly Vertex-Redundantly Rigid Graphs. Graphs and Combinatorics 32, 225–240 (2016). https://doi.org/10.1007/s00373-015-1560-3
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DOI: https://doi.org/10.1007/s00373-015-1560-3