Multigraph augmentation under biconnectivity and general edge-connectivity requirements

Abstract

Given an undirected multigraph G = (V, E) and a requirement function rλ: () → Z+ (where () is the set of all pairs of vertices and Z+ is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the local edge-connectivity and vertex-connectivity between every pair x, y ∈ V become at least rλ(x, y) and two, respectively. In this paper, we show that the problem can be solved in O(n3(m + n) log(n2/(m + n))) time, where n and m are the numbers of vertices and pairs of adjacent vertices in G, respectively. This time complexity can be improved to O((nm + n2 log n) log n), in the case of the uniform requirement rλ(x, y)= 𝓁 for all x, y ∈ V. Furthermore, for the general rλ, we show that the augmentation problem that preserves the simplicity of the resulting graph can be solved in polynomial time for any fixed 𝓁* = max{rλ(x, y) | x, y ∈ V}. © 2001 John Wiley & Sons, Inc.