Augmenting the Edge Connectivity of Planar Straight Line Graphs to Three

We characterize the planar straight line graphs ( Pslg s) that can be augmented to 3-connected and 3-edge-connected Pslg s, respectively. We show that if a Pslg with n vertices can be augmented to a 3-edge-connected Pslg , then at most 2 n −2 new edges are always sufficient and sometimes necessary for the augmentation. If the input Pslg is, in addition, already 2-edge-connected, then n −2 new edges are always sufficient and sometimes necessary for the augmentation to a 3-edge-connected Pslg .