We introduce three new proximity skeletons related to the Voronoi diagram: (1) the Voronoi Graph (VG), which contains the complete symbolic information of the Voronoi diagram without containing any geometry; (2) the Approximate Voronoi Graph (AVG), which deals with degenerate diagrams by collapsing sub-graphs of the VG into single nodes; and (3) the Proximity Structure Diagram (PSD), which enhances the VG with a geometric approximation of Voronoi elements to any desired accuracy. The new skeletons are important for both theoretical and practical reasons. Many applications that extract the proximity information of the object from its Voronoi diagram can use the Voronoi graphs or the PSD instead. In addition, the skeletons can be used as initial structures for a robust and efficient global or local computation of the Voronoi diagram.
We present a space subdivision algorithm to construct the new skeletons, having three main advantages. First, it solves at most uni-variate quartic polynomials. This stands in sharp contrast to previous approaches, which require the solution of a non-linear tri-variate system of equations. Second, the algorithm enables purely local computation of the skeletons in any limited region of interest. Third, the algorithm is simple to implement.