Increasing the vertex-connectivity in directed graphs

Abstract

Given a k-vertex-connected directed graph G, what is the minimum number m, such that G can be made k+1-connected by the addition of m new edges? We prove that if a vertex v has in- and out-degree at least k+1, there exists a splittable pair of edges on v. With the help of this statement, we generalize the basic result of Eswaran and Tarjan, and give lower and upper bounds for m which are equal for k=0 and differ from each other by at most k otherwise. Furthermore, a polynomial approximation algorithm is given for finding an almost optimal augmenting set.