On the hardness of computing maximum self-reduction sequences

Various combinatorial problems on graphs can be approached by reducing the size of the graph according to certain rules. Given an instance of a graph problem, it is desirable to apply a sequence of self-reductions that is as long as possible so that the remaining graph is as small as possible. We show that computing maximum reduction sequences for two self-reduction operations - two-pair reduction and quasi-modular clique reduction - are NP-hard problems, and extend this result to larger sets of self-reductions.