Connectivity augmentation in plane straight line graphs

It is shown that every connected planar straight line graph with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most b(2n − 2)/3c new edges. It is also shown that every planar straight line tree with n ≥ 3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph by adding at most bn/2c edges. These bounds are best possible. However, for every n ≥ 3, there are planar straight line trees with n vertices that do not have an embedding preserving augmentation to a 2-edge connected planar straight line graph by adding fewer than 17 33n−O(1) edges.