Abstract
We consider edge insertion and deletion operations that increase the connectivity of a given planar straight-line graph (PSLG), while minimizing the total edge length of the output. We show that every connected PSLG \(G=(V,E)\) in general position can be augmented to a 2-connected PSLG \((V,E\cup E^+)\) by adding new edges of total Euclidean length \(\Vert E^+\Vert \le 2\Vert E\Vert \), and this bound is the best possible. An optimal edge set \(E^+\) can be computed in \(O(|V|^4)\) time; however the problem becomes NP-hard when G is disconnected. Further, there is a sequence of edge insertions and deletions that transforms a connected PSLG \(G=(V,E)\) into a plane cycle \(G'=(V,E')\) such that \(\Vert E'\Vert \le 2\Vert \mathrm{MST}(V)\Vert \), and the graph remains connected with edge length below \(\Vert E\Vert +\Vert \mathrm{MST}(V)\Vert \) at all stages. These bounds are the best possible.
Research on this paper was supported in part by the NSF awards CCF-1422311 and CCF-1423615. Akitaya was supported by the Science Without Borders program.
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Akitaya, H.A., Inkulu, R., Nichols, T.L., Souvaine, D.L., Tóth, C.D., Winston, C.R. (2017). Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_16
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