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Motivated by applications demand that existing tools could not support, we propose a method for enforcing explicit constraints on a partition’s quotient graph. At the coarsest level of multilevel k-way partitioning, we use several integer linear programming models to bound the graph’s maximum degree, constrain connectivity among groups of subdomains, or match a given reference quotient graph. We present models with linear complexity and investigate additional heuristics. Finally, we show that these constraints improve the performance of a shallow water solver.