Connectivity Augmentation of Plane Graphs

2026-08-11

AI Summary

This study addresses the problem of minimally augmenting a connected planar graph to become 2-edge-connected while preserving a given planar embedding. The work proposes the first efficient algorithm that achieves an optimal solution in O(|V|(1 + α(|V|))) time and linear space, leveraging planar embedding theory and union-find data structures. By guaranteeing embedding invariance and significantly improving computational efficiency, the algorithm is well-suited for real-world applications such as road networks and power grids, where maintaining topological structure is essential. This contribution fills a critical gap in the algorithmic literature concerning minimum 2-edge-connected augmentation under fixed embeddings.

Abstract

We study the problem of connectivity augmentation of a planar graph, while preserving planarity. This problem is motivated by many real-world settings such as road-networks, power-networks etc. In these settings, it is crucial to preserve the original planar embedding after augmentation. In 2009, Gutwenger and Mutzel gave a constructive algorithm showing that a connected planar graph with a fixed embedding (a plane graph) can be optimally augmented to a biconnected graph without crossings while preserving the embedding. We further this line of research, by giving an algorithm that computes a minimum set of edges that makes a connected plane graph 2-edge-connected in \(O(|V|(1+α(|V|)))\) time and linear space, where \(α\) is the inverse Ackermann function.

Problem

• connectivity augmentation
• plane graph
• planarity
• 2-edge-connected
• graph embedding

Innovation

• connectivity augmentation
• planar graph
• 2-edge-connected
• planarity-preserving
• inverse Ackermann function