How do I know that this sudoku puzzle has only solution?

Thinking of sudoku as a graph-coloring problem brings up the question of how one determines that a particular puzzle has a unique solution. If k-coloring a subset of vertices in a graph results in there being only one possible way to k-color the rest we call that subset a k-chromatic forcing set. I will discuss the paper "On the computational complexity of the forcing chromatic number" which looks at how difficult it is to determine the size and the membership of a chromatic forcing set.