Oh and as far as the subgraph goes, sorry, thought that was obvious. Suppose I have a connected graph, and I remove edges until it's disconnected. I now have two connected graphs, both of which are subgraphs of the original graph.

Seva,I'm not sure what the issue is? Of course if you remove enough edges from essentially any graph you will disconnect it, but it is an interesting question to many people to ask, how many edges you have to remove before you are likely to disconnect it. It's obvious of course that if you have a $d$ regular graph, the larger $d$ is the more edges you can remove without disconnecting in general, but I'm asking of $d=3$ I have some ideas about how to maybe consider this, for example by looking at powers of the adjacency matrix, but thought it likely this had been looked at before

Yes, but I'm interested in what happens if you do $4g-3$ cuts initially, thus cutting the original polygon into triangles which can be cut and pasted. Unless I'm misunderstanding you are only doing one initial cut, which severely limits what pasting you can do.

@misha, Yes to the conformal equivalence, but the thing is that "radius" is not meaningless here - think of a short geodesic shrinking to length zero, with "radius" being the distance from the geodesic to the "thick part" (in the sense of Margulis) of the surface. I will however take your advice and rewrite the question to make this clearer, thanks.

Sorry, true, the setting is a surface degenerating to one with a cusp. Harnack would give a bound on norm based on value at one point, since $g$ is continuous on a closed set.