Decomposable graphs and definitions with no quantifier alternation

Let D ( G ) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D 0 ( G ) be the version of D ( G ) where we do not allow quantifier alternations in Φ . Define q 0 ( n ) to be the minimum of D 0 ( G ) over all graphs G of order n . ve that for all n we have log ∗ n − log ∗ log ∗ n − 2 ≤ q 0 ( n ) ≤ log ∗ n + 22 , where log ∗ n is equal to the minimum number of iterations of the binary logarithm needed to bring n to 1 or below. The upper bound is obtained by constructing special graphs with modular decomposition of very small depth.