### Abstract

A.L. Cauchy proved that if the vertex-edge graphs of two convex polyhedra are isomorphic and corresponding faces are congruent then the two polyhedra are the same. This result implies that a convex polyhedron with triangular faces, as a bar-and-joint framework, is rigid. A framework is said to be globally rigid if the bar lengths uniquely determine all pairwise distances between the joints. Global rigidity implies rigidity. It is well-known that every three-dimensional generic bar-and-joint realisation of the graph of a convex polyhedron with triangular faces (that we call a triangulation) is rigid. It is also known that if the number of vertices is at least five then such a realisation of a triangulation is never globally rigid. In this paper we consider braced triangulations, obtained from triangulations by adding a set of extra bars (bracing edges) connecting pairs of non-adjacent vertices. We show that a braced triangulation is generically globally rigid in three-space if and only if it is 4-connected. The special case, when there is only one bracing edge, verifies a conjecture of W. Whiteley. Our proof is based on a new result on the vertex splitting operation which may be of independent interest. We show that every graph which can be obtained from the complete graph on five vertices by non-trivial vertex splitting operations is generically globally rigid in three-space.

Original language | English |
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Pages (from-to) | 249-288 |

Number of pages | 40 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 136 |

DOIs | |

Publication status | Published - May 2019 |

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### Keywords

- Braced triangulation
- Global rigidity
- Rigid graph
- Triangulation
- Vertex splitting

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*136*, 249-288. https://doi.org/10.1016/j.jctb.2018.11.003