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In this second part of a two-part paper, we explore the power and complexity of the g=fKP and g=vKP class of look-ahead operators which can be used to speed up the tree search in the consistent labeling problem. For a specified K and P we show that the fixedpoint power of g=fKP and g=vKP is the same, that g=fKP+1 is at least as powerful as g=fKP, and that g=vK+1p is at least as powerful at g=fKP. Finally, we define a minimal compatibility relation and show how the standard tree search procedure for finding all the consistent labelings is quicker for a minimal relation. This leads to the concept of grading the complexity of compatibility relations according to how much look-ahead work it requires to reduce them to minimal relations and suggests that the reason look-ahead operators, such as Waltz filtering, work so well is that the compatibility relations used in practice are not very complex and are reducible to minimal or near minimal relations by a g=fKP or g=vKP look-ahead operator with small value for parameter P.