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. Further, we prove that there is no local algorithm for augmenting a planar UDG into a 2-edge connected planar graph with straight line edges.

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Keywords 2-edge connected, Augmentation, Deletion, Geometric, Local, Minimum number of edges, Planar, UDG
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Kranakis, E, Krizanc, D. (Danny), Ponce, O.M. (Oscar Morales), & Stacho, L. (Ladislav). (2010). Bounded length, 2-edge augmentation of geometric planar graphs. doi:10.1007/978-3-642-17458-2_31