Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs (1612.04780v1)

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.