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A graph is said to be (k, l)-connected if the resulting graph after the removal of any k vertices and \((l-1)\) edges or the removal of any \((k-1)\) vertices and l edges is still connected. Beineke and Harary (Mathematika 14:197–202, 1967) 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 is (k, l)-connected. However, Mader (Surveys in combinatorics, London mathematical society lecture note series, vol 38, pp 66–95, 1979) pointed out a gap in this proof. In this paper, we first modify the conclusion (by changing from k to \(k+1\) vertex-disjoint paths), 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 formulations and a cutting plane algorithm. Numerical experiments are performed on randomly generated graphs to compare these approaches.

RETRACTED ARTICLE: On the survivable network design problem with mixed connectivity requirements