  Given a graph $G$, we define ${\bf bcg}(G)$ as the minimum $k$ for which $G$
can be contracted to the uniformly triangulated grid $\Gamma_{k}$. A graph
class ${\cal G}$ has the SQG${\bf C}$ property if every graph $G\in{\cal G}$
has treewidth $\mathcal{O}({\bf bcg}(G)^{c})$ for some $1\leq c<2$. The
SQG${\bf C}$ 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
SQG${\bf C}$ property. This family includes, in particular, bounded-degree
string graphs. This considerably extends the applicability of bidimensionality
theory for contraction bidimensional problems.
