Bounded Length, 2-Edge Augmentation of Geometric Planar Graphs

Algorithms for the construction of spanning planar subgraphs of Unit Disk Graphs (UDGs) do not ensure connectivity of the resulting graph under single edge deletion. To overcome this deficiency, in this paper we address the problem of augmenting the edge set of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We give bounds on the number of newly added straight-line edges and show that such edges can be of length at most 3 times the max length of the edges of the original graph; also 3 is shown to be optimal. It is shown to be NP-hard to augment a geometric planar graph to a 2-edge connected geometric planar with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. We prove that this problem cannot be solved locally if the nodes do not know their coordinates.