The Price of Connectivity Augmentation on Planar Graphs

Hugo A. Akitaya; Justin Dallant; Erik D. Demaine; Michael Kaufmann; Linda Kleist; Frederick Stock; Csaba D. Tóth; Torsten Ueckerdt

arXiv · 2509.01096

The Price of Connectivity Augmentation on Planar Graphs

Hugo A. Akitaya·Justin Dallant·Erik D. Demaine·Michael Kaufmann·Linda Kleist·Frederick Stock·Csaba D. Tóth·Torsten Ueckerdt

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Abstract

Given two classes of graphs, $\mathcal{G}_1\subseteq \mathcal{G}_2$, and a $c$-connected graph $G\in \mathcal{G}_1$, we wish to augment $G$ with a smallest cardinality set of new edges $F$ to obtain a $k$-connected graph $G'=(V,E\cup F) \in \mathcal{G}_2$. In general, this is the $c\to k$ 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 $c\to k$ augmentation problem is NP-complete when $2\leq c<k\leq 5$. 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 $c\to k$ connectivity augmentation problem for arbitrary $k\in \mathbb{N}$, 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.

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Hugo A. Akitaya, Justin Dallant, Erik D. Demaine, Michael Kaufmann, Linda Kleist, Frederick Stock, Csaba D. Tóth, Torsten Ueckerdt. 2025-09-01. The Price of Connectivity Augmentation on Planar Graphs. https://arxiv.org/abs/2509.01096

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