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The following Python program runs in 6.4s (under PyPy).

As noted in the comments in the code, in the initial stages of the computation it can be deduced analytically what the solution to the puzzle is. But we want to see a constructive solution, so the program goes on to generate all possible sets of dominoes that satisfy the conditions of the puzzle, and verify that they all correspond to the solution.

My original Perl code ran in 2m6s.

Solution:The lost dominoes are 0-0 (double blank) and 2-2 (double two).