Connectivity augmentation in planar 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 ⌊ ( 2 n − 2 ) / 3 ⌋ 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 with at most ⌊ n / 2 ⌋ new edges. These bounds are the 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 with fewer than 17 33 n − O ( 1 ) new edges.