subdividing at most one edge in each of $C_1$ and $C_2$. We say that such a graph $G'$ is a *near $3,3$-quadrangulation*.

We say that a graph $G$ embedded in the cylinder is *tame* if $G$ contains no contractible triangles, and all triangles of $G$ are pairwise vertex-disjoint. Let $G$ be a tame graph embedded in the cylinder with rings of length at most $4$. We say that $G$ is a *chain* of graphs $G_1$, …, $G_n$, if there exist non-contractible $(\le\!4)$-cycles $C_0$, …, $C_n$ in $G$ such that

-   the cycles are pairwise vertex-disjoint except that for $(i,j) \in \{(0,1),(n,n-1)\}$, $C_i$ can intersect $C_j$ if $C_i$ is a 4-cycle and $C_j$ is a triangle,

-   for $0\le i<j<k\le n$ the cycle $C_j$ separates $C_i$ from $C_k$,

-   the cycles $C_0$ and $C_n$ are the rings of $G$,

-   every triangle of $G$ is equal to one of $C_0$, …, $C_n$, and

-   for $1\le i\le n$, the subgraph of $G$ drawn between $C_{i-1}$ and $C_i$ is isomorphic to $G_i$.

We say that $C_0$, …, $C_n$ are the *cutting cycles* of the chain. The main result of this paper is the following.

\[thm-many\] There exists an integer $c\ge 0$ such 