There is a related question on how to generate all regular graphs; however, the procedure is random and repeats the generated graphs. Plus, there is no stop condition, unless recording the total number of non-isomorphic graphs and matching it to the known numbers.

I wonder if there is a *more efficient procedure* that loops over all possible regular/bidegreed graphs? By more efficient, I mean return each regular graph only once, but if it's hard maybe other generation schemes that minimize the number of repetitions.

Ideally, I would like to have this procedure to generate all bidegreed graphs of order 20-30. But if bidegreed graphs are tough, at least for regular graphs.

By bidegreed graphs, I mean connected graphs that have only two possible values of degrees, $a$ and $a+1$, for any $a \ge 2$. The motivation for this is the reconstruction conjecture for this type of graphs.