Data strucutre for finite graphs on a cylinder
Let the vertices at the top be $a_0, \dots, a_{n-1}$ and those at the bottom be $b_0,\dots, b_{n-1}$, both ordered, say, clockwise. If the graph has no edge-crossings, the edge set can only be $\{(a_i,b_{i+k})\mid 0\leq i\leq n\}$ for some $k$, with addition carried out modulo&nbsp;$n$. So the edge set is uniquely determined by the value of&nbsp;$k$.

Beyond that, though, it's not clear what you're asking. The embedding of the graph in the surface of the cylinder could be much more complicated than you're suggesting. For example, the edges could all spiral anticlockwise around the cylinder thirteen times. Or they could spiral anticlockwise three times, then clockwise four times, and then wiggle around lots. Perhaps you have a requirement that the edges can't spiral multiple times? And/or that they're straight lines if you "unwrap" the cylinder and lay it flat?  If so, note that gluing two cylinders together doesn't maintain this property: for example, if you glue the cylinder that has edges $(a_i,b_{i+1})$ using shortest paths to the cylinder that has edges $(a_i,b_{i-1})$ using shortest paths, you get a cylinder whose edges zig-zag, so aren't straight lines, don't use shortest paths and don't "go the long way around". Likewise, if you glue $2n$ copies of the cylinder that uses shortest paths to link $a_i$ to $a_{i+1}$, you get a cylinder whose edges spiral twice.