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+FG+F is (k+1)(k+1)-connected. The complexity status of this problem is an open question. The problem admits a 2-approximation algorithm. Another algorithm due to Jordán computes an augmenting edge set with at most ⌈(k−1)/2⌉⌈(k−1)/2⌉ edges over the optimum. C⊂V(G)C⊂V(G) is a k-separator (k-shredder) of G   if |C|=k|C|=k and the number b(C)b(C) of connected components of G−CG−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+1b(C)⩾k+1. This leads to a new splitting-off 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⩾3p⩾3) is less than 2n/(2p−3)2n/(2p−3), and that this bound is asymptotically tight.