On shredders and vertex connectivity augmentation

We consider the following problem: given a k-(node) connected graph G find a smallest set F of new edges so that the graph G + F is (k + 1)-connected. The complexity status of this problem is an open question. The problem admits a 2approximation algorithm. Another algorithm due to Jordán computes an augmenting edge set with at most d(k− 1)/2e edges over the optimum. C ⊂ V (G) is a k-separator (k-shredder) of G if |C| = k and the number b(C) of connected components of G− C is at least two (at least three). We will show that the problem is polynomially solvable for graphs that have a k-separator C with b(C) ≥ k+1. This leads to a new splittingoff theorem for node connectivity. We also prove that in a k-connected graph G on n nodes the number of k-shredders with at least p components (p ≥ 3) is less than 2n/(2p− 3), and that this bound is asymptotically tight.