are drawn in Figure \[fig-thomaswalls\].

![Some Thomas-Walls graphs.[]{data-label="fig-thomaswalls"}](fig-thomaswalls)

For $n\ge 2$, note that $T_n$ contains unique $4$-cycles $C_1=u_1u_2u_3u_4$ and $C_2=v_1v_2v_3v_4$ such that $u_1u_3,v_1v_3\in E(G)$. Let $T'_n=T_n-\{u_1u_3,v_1v_3\}$. We also define $T'_1$ to be a $4$-cycle $C_1=C_2=u_1v_1u_3v_3$. We call the graphs $T'_1$, $T'_2$, …*reduced Thomas-Walls graphs*, and we say that $u_1u_3$ and $v_1v_3$ are their *interface pairs*. Note that $T'_n$ has an embedding in the cylinder with rings $C_1$ and $C_2$.

A *patch* is a graph $F$ drawn in the disk with ring $C$ of length $6$, such that $C$ is an induced cycle in $F$ and every face of $F$ other than the one bounded by $C$ has length $4$ and every 4-cycle bounds a face. Let $G$ be a graph embedded in the sphere, possibly with holes. Let $G'$ be any graph which can be obtained from $G$ as follows. Let $S$ be an independent set in $G$ such that every vertex of $S$ has degree $3$. For each vertex $v\in S$ with neighbors $x$, $y$ and $z$, remove $v$, add new vertices $a$, $b$ and $c$ and a $6$-cycle $C=xaybzc$, and draw any patch with ring $C$ in the disk bounded by $C$. We say that any such graph 