I'm trying to list the connected 3-sheeted covering spaces of $S^1 \vee S^1$ up to isomorphism. I know what the answer is; I've seen listings.
My question is, if I'm deriving this, how do I know I've got all the covers?
Is there a nice combinatorial or graph theoretic way to count the number of covers or systematically exhaust the possibilities? Any such cover is a two-oriented connected graph with three vertices... is there a nice way to count these?
I've seen that this question was asked before, but the answers didn't address the issue of making sure all covers are found. Thanks.