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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.)

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 in $E^2$.)

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.)

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.)

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.

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.)