Minimum weight connectivity augmentation for planar straight-line graphs

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 ∪ E+) by adding new edges of total Euclidean length ║E+║ ≤ 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 plane cycle G′ = (V,E′) such that ║E′║ ≤ 2║MST(V)║, and the graph remains connected with edge length below ║E║+║MST(V)║ at all stages. These bounds are the best possible.