On the Survivable Network Design Problem with Mixed Connectivity Requirements

A graph is said to be (k, l)-connected if the resulted graph after removing of any k vertices and (l−1) edges or removing of any (k−1) vertices and l edges is still connected. Beineke and Harary (1967) (see [1]) claimed to prove that there should be k+ l edge-disjoint paths, of which k are vertex-disjoint, between any pair of vertices if the graph has (k, l)-connectivity. However, Mader (1979) (see [2]) pointed out a gap in this proof. In this paper, we first modify the conclusion (by changing to k+1 vertex-disjoint paths instead of k), and then formally prove it. As an application, we propose to design a (k, l)-connected network with minimum cost, by presenting two integer programming (IP) formulations and a cutting plane algorithm. Numerical experiments are performed on randomly generated graphs to compare these approaches.