On depth-reduction and grates

For each ε(0 ≤ ε ≪ 1) a family Gn =(V(Gn), E(Gn)) of a cyclic digraphs can be constructively defined having the following properties: (a) #V(Gn) ≤ n ¿ 2n+2 (b) degree (Gn) ≤ constant (c) it is necessary to remove Ω(n ¿ 2n) edges in order to reduce the depth of Gn to (2n)ε. It is then shown: For suitable constants c1, C2 ≫ 0, there are (fn, n)- grates (see Definition 1) of size linear in n, where fn(x):= c1 ¿ n2 x ≤ c2 ¿ n/0 otherwise