Cubic Augmentation of Planar Graphs

TL;DR

This paper investigates the complexity of augmenting planar graphs to become 3-regular while maintaining planarity, providing efficient algorithms for certain cases and proving NP-hardness for others.

Contribution

It introduces algorithms for fixed-embedding cases and proves NP-hardness for general augmentation and triconnected augmentation scenarios.

Findings

Efficient algorithm for fixed-embedding 3-regular planar augmentation.

NP-hardness results for general augmentation and triconnected augmentation.

Generalization to connected and biconnected augmentation cases.

Abstract

In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm for the variant of the problem where the input graph has a fixed planar (topological) embedding that has to be preserved by the augmentation. We further generalize this algorithm to test efficiently whether a 3-regular planar augmentation exists that additionally makes the input graph connected or biconnected. If the input graph should become even triconnected, we show that the existence of a 3-regular planar augmentation is again NP-hard to decide.