Before revision:  The class G is the family of quadrangulations obtained from the 4-cycle C_4 by successively inserting a single vertex or a 4-cycle of vertices into a face  .
After revision:  The class G is the family of quadrangulations obtained from the 4-cycle C_4 by successively inserting a single vertex or a 4-cycle of vertices into a face . For every graph G\in \mathcal{G .