Hi axiel,
Sudoku solving algorithms in Wikipedia is a low quality article that can only confuse you.
You can read
There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem by
Gary McGuire and others.
Try to understand Unavoidable Sets (UA) concept. You may skip details for reduction of the number of grids for investigation, parallelism, etc.
Authors provide an excellent algorithm (and code) for determining the short-sized UA. I have implemented it in gridchecker, if you find there it is more readable.
From your initial post I am unsure whether you are interested in minimal puzzles which givens are subset of a non-minimal puzzle, or you want minimal puzzles for particular solution grid regardless of the givens in your initial puzzle.
For the first case, you can apply McGuire's algorithm with additional constraint that givens shouldn't be in a blacklisted cells which you compose form the non-givens of your initial puzzle. The blacklisted cells also reduce the sizes of UA, sometimes to 1 which in fact forces a particular cell as a given.
The effectivness of McGuire's algorithm decreases with the size of the searched puzzles. Gridchecker has code with some experimental "improvements" for searchin puzzles of larger sizes. This complicates the code, likely unnecessary in some parts, making it difficult to understand.
Gridchecker also has code for reduction of givens that is useful for larger puzzles. At its limit, the top-down approach takes forever when started from a full solution grid, but generates some high-clue minimal puzzles in a reasonable time.
Gridchecker still keeps the initial implementation for determining the small sized puzzles within a fixed solution grid - the one before McGuire's improvements. It was effective for its time, but was overperformed in several orders of magnitude later. It attempts to determine a "dense" area of givens in the solution grid and exploits this knowledge for earlier cancellation of impossible clue placement paths. Likely this code is only in historical interest and its benefits are difficult/impossible to incorporate in other alogorithms.
For low-clue determination of a grid, see
this topic by
Mathimagics.
For exhaustive 17-clue puzzles search, there is an
ongoing project lead by
champagne.
Enjoy,
M.