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Demonstrating the lower bound. (A) A contraction of a branch with a free-child can only possibly result in non-trivial adjacencies becoming trivial. (B) An adjacency deletion can at most reduce the number of modules by 2, and if the number of modules decrease by two then the number of unattached sides in a modules decreases by 2. (C) An example of the deletion of an adjacency redistributing two unattached sides.
