Minimum Weight Connectivity Augmentation for Planar Straight-Line Graphs

TL;DR

This paper studies how to augment planar straight-line graphs to improve connectivity with minimal total edge length, providing bounds, algorithms, and complexity results for edge insertion and deletion operations.

Contribution

It introduces optimal bounds for augmenting connectivity in PSLGs, including a polynomial-time algorithm for 2-connectivity augmentation and NP-hardness results for disconnected graphs.

Findings

Augmentation to 2-connected PSLGs with total added length ≤ 2× original edges.

NP-hardness of optimal augmentation for disconnected PSLGs.

Existence of edge sequences transforming PSLGs into cycles with bounded total length.

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.