$G'$ is obtained from $G$ by *patching*. This operation was introduced by Borodin et al. [@4c4t] in the context of describing planar $4$-critical graphs with exactly $4$ triangles.

Consider a reduced Thomas-Walls graph $G=T'_n$ for some $n\ge 1$, with interface pairs $u_1u_3$ and $v_1v_3$. A *patched Thomas-Walls graph* is any graph obtained from such a graph $G$ by patching, and $u_1u_3$ and $v_1v_3$ are its interface pairs (note that $u_1$, $u_3$, $v_1$, and $v_3$ have degree two in $G$, and thus they are not affected by patching).

Let $G$ be a graph embedded in the sphere with $n$ holes ($n\in\{1,2\}$), with rings $C_i=x_iy_iz_iw_i$ of length $4$, for $1\le i\le n$. Let $y'_i$ be either a new vertex or $y_i$, and let $w'_i$ be either a new vertex or $w_i$. Let $G'$ be obtained from $G$ by adding $4$-cycles $x_iy'_iz_iw'_i$ forming the new rings. We say that $G'$ is obtained by *framing on pairs $x_1z_1$, …, $x_nz_n$*.

Let $G$ be a graph embedded in the cylinder with rings $C_1$ and $C_2$ of length three, such that every other face of $G$ has length $4$. We say that such a graph $G$ is a *$3,3$-quadrangulation*. Let $G'$ be obtained from $G$ by 