The question is in the title! I know that a globally rigid graph is 3-connected and redundantly rigid, so my question could be rephrased as: "does there exist a graph which is 3-connected and chordal but not redundantly rigid?"


It seems fairly intuitive to me that there does not, but my intuition about graphs has a fairly bad record...


Thanks in advance.



Does there exist a 3-connected, chordal graph which is not globally rigid?
Every such graph is generically globally rigid in $E^2$. A 3-connected chordal graph can be built by starting with a triangle and then sequentially attaching new vertices to at least 3 previous ones. See [this paper](http://arxiv.org/abs/1205.3990) for some explicit statements. This idea generalizes to any dimension.


In fact, any generic (or even general position) framework for such a graph will be universally rigid in $E^2$, ie. it has no equivalent and non-congruent frameworks in **any** dimension. Such a graph is called generically universally rigid in $E^2$. (Note that you need to be a bit careful with universal rigidity as there are graphs that have some generic frameworks that are universally rigid in $E^2$, and other generic frameworks that are not universally rigid in $E^2$.)


