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We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let μ be a function on the subsets of vertices of a graph G. In the (μ,p,q)-PARTITION problem, the task is to find a partition of the vertices into clusters where each cluster C satisfies the requirements that (1) at most q edges leave C and (2) μ(C)< p. Our first result shows that if μ is an arbitrary polynomial-time computable monotone function, then (μ,p,q)-PARTITION can bearXiv:1711.03885v1 fatcat:7ey2vvvomfaalhbiibu3kzdh7e