Let A be a structure and H a partial graph with V(H) ⊆ V(A). Then H is definable in A by formulas of quantifier rank q₀ with parameters from a set S ⊆ V(A) if for every relation R of H of arity k there is a formula φ_R(x₁, …, x_k) of quantifier rank at most q₀ with parameters from S such that, for all v̄ ∈ V(A)^k,

A ⊨ φ_R(v̄) ⟺ v̄ ∈ V(H)^k and H ⊨ R(v̄).

Let G be a graph and r, c, s, q₀ ∈ ℕ. A construction sequence G₁, …, G_λ of G is radius-r definable with c unary predicates, s parameters and quantifier rank q₀ if, for all t ∈ [λ−1] and v ∈ V(G), the induced partial graph G_t[Ball^{G_{t+1}}_r(v)] is definable in some expansion of G by c unary predicates, by formulas of quantifier rank q₀ with parameters from a set of at most s vertices. The expansion and the parameters may depend on t and v.