Using your Head is Permitted

Briefly, it asks the following question: if I am trying to solve an n-piece jigsaw puzzle, and the only tool at my disposal is the ability to determine whether two given tiles will fit together in the final construction, how many tile matches do I need to check in order to solve the puzzle?

To make this question more concrete, the following model was offered: the puzzle contains a known set of n positions, each of which must be fitted with exactly one tile, and it is known which positions are adjacent to which other positions. That is, the shape of the constructed jigsaw puzzle is known, describable as a connectivity graph over the positions. (Most of the world's graphs are "canonical", meaning that their connectivity graph is a grid, but this question allows any other connectivity graph as well.) Let d be the degree of the connectivity graph.

We assume that the connectivity graph has no self-symmetries, meaning that if one was to compare every pair of tiles to see if they are to be mapped to neighbouring positions on the connectivity graph, one will have excluded by this all possible assignments of tiles to positions, except for the one true solution to the puzzle.

Define the "difficulty" of a puzzle to be the minimum number of pairs of tiles that one must try to match together in this puzzle before one is sure to have narrowed down the list of possible assignments to only the one true solution. By "minimum" we mean that one employs the best possible strategy for the given graph, and by "sure" we mean that one must take into account the worst possible sequence of outcomes to the tile-match operations.

Define the difficulty of a set of puzzles with unboundedly increasing n to be the complexity of the difficulty of the set's individual puzzles, as a function of n.

The question was: let S be such a set of puzzles with unboundedly increasing n and bounded d, such that S's difficulty is the minimum possible complexity for any such set. What is this complexity?

This month's riddle:

Solve this riddle again, only this time, do not restrict yourself to sets of puzzles with a bounded d.

All answers should come with proofs, as usual. Solvers are welcomed to check out the answer to the original riddle.

List of solvers: