Is there a name for this type of quadrangulations

Question

A quadrangulation is a planar graph such that every face is bounded by a 4-cycle. The quadrangulations below are something I have encountered, where it recursively replaces each innermost 4-cycle with a cube, as shown in the figure. Does such graph class have a name? Is there a concise representation of them?

Answer 1

Mathematica treats these as a subclass of the $3$-dimensional grid graphs: the $n^{\text{th}}$ graph in this sequence is the $2 \times 2 \times n$ grid graph. So, for example, either GraphData[{"Grid", {2, 2, 5}}] or GridGraph[{2, 2, 5}] would produce a $20$-vertex quadrangulation of this type (though only the GraphData version will display the embedding you want by default).

Another way to put it is: the $n^{\text{th}}$ quadrangulation in the sequence can be written as $P_2 \mathbin{\square} P_2 \mathbin{\square} P_n$ or as $C_4 \mathbin{\square} P_n$, where $\square$ denotes the Cartesian product and $P_n$ denotes the $n$-vertex path.