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For every integer k 2 and graph G, consider the following natural procedure: if G has a component G that is not k-connected, remove G if |G | k, otherwise remove a cutset U ⊂ V (G ) with |U | < k; do the same with the remaining graph until only k-connected components are left or all vertices are removed. We are interested when this procedure stops after removing o(|G|) vertices. Surprisingly, for every graph G of order n with minimum degree (G) √ 2(k − 1)n, the procedure always stops afterdoi:10.1016/j.dam.2006.07.007 fatcat:ow6lukeq4rdmjavlrohr4tnu4m