Compatible Connectivity-Augmentation of Planar Disconnected Graphs

TL;DR

This paper addresses the problem of augmenting disconnected planar graphs to become connected while maintaining planarity and isomorphism across multiple drawings, providing tight bounds and efficient algorithms.

Contribution

It establishes tight bounds and efficient algorithms for compatible connectivity-augmentation of planar graphs across multiple isomorphic drawings.

Findings

Adding (nr^{1-1/k}) edges and vertices suffices and is sometimes necessary.

The augmentation can be computed in (nr^{1-1/k}) time for fixed k.

The bounds are tight for various ranges of r and k.

Abstract

Motivated by applications to graph morphing, we consider the following \emph{compatible connectivity-augmentation problem}: We are given a labelled $n$-vertex planar graph, $\mathcal{G}$, that has $r\ge 2$ connected components, and $k\ge 2$ isomorphic planar straight-line drawings, $G_1,\ldots,G_k$, of $\mathcal{G}$. We wish to augment $\mathcal G$ by adding vertices and edges to make it connected in such a way that these vertices and edges can be added to $G_1,\ldots,G_k$ as points and straight-line segments, respectively, to obtain $k$ planar straight-line drawings isomorphic to the augmentation of $\mathcal G$. We show that adding $\Theta(nr^{1-1/k})$ edges and vertices to $\mathcal{G}$ is always sufficient and sometimes necessary to achieve this goal. The upper bound holds for all $r\in\{2,\ldots,n\}$ and $k\ge 2$ and is achievable by an algorithm whose running time is $O(nr^{1-1/k})$ for $k=O(1)$ and whose running time is $O(kn^2)$ for general values of $k$. The lower bound holds for all $r\in\{2,\ldots,n/4\}$ and $k\ge 2$.