Augmenting the Connectivity of Planar and Geometric Graphs

Ignaz Rutter,Alexander Wolff

doi:10.7155/jgaa.00275

Abstract

In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s{t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution.

Highlights

Augmenting a given graph to increase its connectivity is important, e.g., for securing communication networks against node and link failures
We show that it is NP-hard to find a minimumcardinality augmentation that makes a planar graph 2-edge connected
We further show that both problems are hard in the geometric setting, even when restricted to trees

Summary

Introduction

Augmenting a given graph to increase its connectivity is important, e.g., for securing communication networks against node and link failures. Given a planar graph G = (V, EG) and a planar biconnected (bridge-connected) graph H = (V, EH ) with EG ⊆ EH , find a smallest set E ⊆ EH such that G = (V, EG ∪ E ) is planar and biconnected (bridgeconnected) They show that both problems are NPhard if G is not necessarily connected and give O(n4)time algorithms for the connected cases. Given a geometric graph G we again want to find a (small) set of vertex pairs such that adding the corresponding edges to G leaves G plane and augments its connectivity. Given a plane geometric graph G, two vertices s and t of G, and an integer k > 0, is it possible to augment G such that it contains k edge-disjoint (k vertex-disjoint) s–t paths?

Complexity results

Convex geometric graphs