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 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.)rigid in $E^2$.)

4 Defined universal rigidity.

Every such graph is generically globally rigid. 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 for some explicit statements.

In fact, any generic (or even general position) framework for such a graph will be universally rigid, ie. it has no equivalent and non-congruent frameworks in any dimension. Such a graph is called generically universally rigid. (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, and other generic frameworks that are not universally rigid.)

3 added 262 characters in body

Every such graph is generically globally rigid. 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 for some explicit statements. In fact, any generic (or even general position) framework for such a graph will be universally rigid. Such a graph is called generically universally rigid. (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, and other generic frameworks that are not universally rigid.)