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 D₀ (G) be the version of D (G) where we do not allow quantifier alternations in Φ. Define q₀ (n) to be the minimum of D₀ (G) over all graphs G of order n. We prove that for all n we have log^* n - log^* log^* n - 2 ≤ q₀ (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.