Augmenting a (K — 1)-vertex-connected multigraph to an ℓ-edge-connected and k-vertex-connected multigraph

Abstract

Given an undirected multigraph G = (V, E) and two positive integers ℓ and k, we consider the problem of augmenting G by the smallest number of new edges to obtain an ℓ-edge-connected and k-vertex-connected multigraph. In this paper, we show that an (k — 1)-vertex-connected multigraph G (k ≥ 4) can be made ℓ-edge-connected and k-vertex-connected by adding at most 2ℓ surplus edges over the optimum, in 0(min{k, √n}kn3 + n4) time, where n = |V|.