Generating many graphs satisfying some properties Suppose I want to generate all graphs satisfying some properties (specifically 2-connected graphs on 30 vertices with degree sequence $(4, 4, 3, 3,3, \dotsc, 3)$). Unfortunately, there are an enormous number of such graphs. Probably the fastest way to find all of them would to use nauty, but it is too slow.

Is there a way to just generate many such graphs (i.e., not needing a complete list)? 

There isn't an obvious way to get nauty to output just the first 10 million graphs, for instance. Maybe this isn't even possible, I don't know how nauty works.
There are methods for randomly sampling graphs with a given degree sequence. You could try sampling and post-selecting for 2-connectedness, since that test is essentially a DFS. See for example

Efficient and Exact Sampling of Simple Graphs with Given Arbitrary Degree Sequence