I guess this is not the right place to ask about sudoku variants, but then again, I'm looking for generic rules of using uniqueness methods. I've never actually used them on other than plain vanilla sudokus, as I've only solved basic Killers, Sudoku Xs, and (only a few) Jigsaws so far, but started thinking about it the other day. Even when solving basic puzzles, URs etc can be really handy because they're often available and almost always easier to spot than triples and quads, especially if solving without pencil marks. Quite often they can be used to bypass such harder parts of basic puzzles, and I freely do that because I avoid using pencil marks with basic(+) puzzles as much as possible. A couple of times I've even spotted extended (2x3) URs without pencil marks, which solved those (non-basic) puzzles quite easily. BUGs, of course, are usually even easier.

Now, my question is about URs (and their generalizations) on some sudoku variants. It's quite obvious that the plain vanilla rules (four corners, two boxes) aren't enough because of the additional constraints. A quick-and-dirty analysis would suggest that a UR on a Killer puzzle would only work if the corners are not only in two boxes but two cages as well. Similarly I would imagine that a UR would only work on a Sudoku X if none of the corners is on a diagonal. Am I on the right track so far? I can't see similarly obvious restrictions on Jigsaws (no additional constraint sets), but I'm not very familiar with them anyway so I might easily miss something.

How would you generalize those observations (assuming they were correct) for any sudoku variant? What about BUGs and other larger DP structures? I would imagine that using them safely might be much harder than simple URs if additional constraint sets are in play.