On the Optimal Vertex-Connectivity Augmentation

This paper considers the problem of finding a minimum-cardinality set of edges for a given k-connected graph whose addition makes it (k + 1)-connected. We give sharp lower and upper bounds for this minimum, where the gap between them is at most k − 2. This result is a generalization of the solved cases k = 1, 2, where the exact min-max formula is known. We present a polynomial-time approximation algorithm which makes a k-connected graph (k + 1)-connected by adding a new set of edges with size at most k − 2 over the optimum.