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We characterize the 2-connected and 2-edge connected planar straight line graphs, respectively , that have embedding preserving augmentations to 3-connected and 3-edge connected planar straight line graphs. If such an augmentation is possible, then it can be done using at most n − 2 new edges in the worst case. These bounds are best possible.
It is shown that if a planar straight line graph (PSLG) with n vertices in general position in the plane can be augmented to a 3-edge-connected PSLG, then 2n−2 new edges are enough for the augmentation. This bound is tight: there are PSLGs with n ≥ 4 vertices such that any augmentation to a 3-edge-connected PSLG requires 2n − 2 new edges.