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 2-approximation algorithm. Another algorithm due to Jordán computes an augmenting edge set with at most ⌈ (k - 1) / 2 ⌉ 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 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 ≥ 3) is less than 2 n / (2 p - 3), and that this bound is asymptotically tight.