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arXiv · 1612.04780

Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

Hugo A. Akitaya·Rajasekhar Inkulu·Torrie L. Nichols·Diane L. Souvaine·Csaba D. Tóth·Charles R. Winston
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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 $\|E^+\|\leq 2\|E\|$, 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 planar straight-line cycle $G'=(V,E')$ such that $\|E'\|\leq 2\|{\rm MST}(V)\|$, and the graph remains connected with edge length below $\|E\|+\|{\rm MST}(V)\|$ at all stages. These bounds are the best possible.

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Hugo A. Akitaya, Rajasekhar Inkulu, Torrie L. Nichols, Diane L. Souvaine, Csaba D. Tóth, Charles R. Winston. 2016-12-14. Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs. https://arxiv.org/abs/1612.04780

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