The Price of Connectivity Augmentation on Planar Graphs

2025-08-31

AI Summary

This work addresses the minimum edge-augmentation problem for transforming $c$-connected graphs into $k$-connected graphs within the class of $ell$-planar graphs (including planar graphs). We propose the first unified modeling framework for arbitrary $2 leq c < k leq 5$ and $ell geq 0$, establishing tight trade-off bounds between connectivity $k$ and local crossing number $ell$. Crucially, we establish a deep connection between connectivity augmentation and triangulation edge-flip distance—resolving the long-standing open problem (posed in 2014) of computing the minimum flip distance to a $4$-connected triangulation. We prove NP-completeness of this problem on planar graphs and design an EPTAS for augmenting triangulations to $4$-connected triangulations. Our approach integrates computational geometry, topological embedding analysis, and edge-flip techniques, extending beyond classical planarity constraints to advance the theoretical foundations and algorithmic design of highly connected structures in crossing-bounded graph classes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Given two classes of graphs, $mathcal{G}_1subseteq mathcal{G}_2$, and a $c$-connected graph $Gin mathcal{G}_1$, we wish to augment $G$ with a smallest cardinality set of new edges $F$ to obtain a $k$-connected graph $G'=(V,Ecup F) in mathcal{G}_2$. In general, this is the $c o k$ connectivity augmentation problem. Previous research considered variants where $mathcal{G}_1=mathcal{G}_2$ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the $c o k$ augmentation problem is NP-complete when $2leq c<kleq 5$. However, the connectivity of the augmented graph $G'$ is at most $5$ if $mathcal{G}_2$ is limited to planar graphs. We initiate the study of the $c o k$ connectivity augmentation problem for arbitrary $kin mathbb{N}$, where $mathcal{G}_1$ is the class of planar graphs, plane graphs, or planar straight-line graphs, and $mathcal{G}_2$ is a beyond-planar class of graphs: $ell$-planar, $ell$-plane topological, or $ell$-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity $k$ and the local crossing number $ell$ of the augmented graph $G'$. We also show that our hardness results apply to this setting. The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.

Problem

• Augmenting planar graphs to higher connectivity
• NP-completeness of connectivity augmentation problems
• Tradeoffs between connectivity and crossing number

Innovation

• Augment planar graphs with minimal edges
• NP-complete for 2 to 5 connectivity augmentation
• Tight bounds on connectivity and crossing number