Making the components of a graph kk-connected

For every integer k⩾2k⩾2 and graph GG, consider the following natural procedure: if GG has a component G′G′ that is not kk-connected, remove G′G′ if |G′|⩽k|G′|⩽k, otherwise remove a cutset U⊂V(G′)U⊂V(G′) with |U|