Making the components of a graph k-connected

For every integer k 2 and graph G; consider the following natural procedure: if G has a component G0 that is not k-connected, remove G0 if jG0j k, otherwise remove a cutset U V (G0) with jU j < 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 (jGj) vertices. Surprisingly, for every graph G of order n with minimum degree (G) p 2 (k 1)n; the procedure always stops after removing at most 2n (k 1) = vertices. We give examples showing that our bounds are essentially best possible.