I have a graph and I'm trying to enumerate all paths that start at node $S$, end at node $E$, uses each edge, and minimizes the number of edges used (i.e. it can only re-use an edge if strictly necessary).
What I am wondering is that if I have a solution path, e.g. $SADBCABE$, can I find all other valid paths that use the same set of edges by iterating over the ways in which I can reverse loops?
In other words, the example path has two loops: one that over $A$ ($ADBCA$), and one over $B$ ($BCAB$). This leads to five possible paths that can be made by reversing these paths that can be named: $A_0B_0$, $A_0B_1$, $A_1B_0$, $A_1B_1$, and $B_1A_1$. Here, $B_1A_1$ refers to reversing path $B$ and then reversing path $A$. This example makes it clear that order matters.
Will this method of enumeration miss any paths containing the same edge set?
EDIT: After some discussion in the comments, it is clear that further reversals on the five paths listed above could produce previously unfound paths. It seems like to use this methodology, I would also have to keep a fringe set and ensure that all reversals on the fringe produce paths that I've already found before removing it. This would no longer be a simple enumeration of permutations, but rather a BFS, which is unfortunately less deterministic. Is this already a known problem that I could read more about? I've tried Googling various things about domino trains (since each edge could be thought of as a domino), but I didn't find anything.