Combinatorial pseudo-triangulations

We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of [Haas et al. Planar minimally rigid graphs and pseudo-triangulations, Comput. Geom. 31(1-2) (2005) 31-61] which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain ''generalized Laman property'' is a necessary and sufficient condition for a CPT to be ''stretchable'', and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of CPTs on closed surfaces.

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