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Given a graph G, we define bcg (G) as the minimum k for which G can be contracted to the uniformly triangulated grid Γ k . A graph class G has the SQGC property if every graph G ∈ G has treewidth O (bcg (G) c) for some 1 ≤ c < 2 . The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQGC property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems. [ABSTRACT FROM AUTHOR]