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In the (unweighted) minimum k-cut problem, the goal is to partition the nodes in a given graph to at least $k$ subsets, such that the number of edges between different subsets is as small as possible.

I am interested in a variant in which the number of subsets is not fixed in advance, but there can be at most $t$ vertices in each subset. So the goal is to partition the nodes into subsets of cardinality at most $t$, such that the number of edges between different subsets is as small as possible.

Is there a polynomial-time algorithm for this problem, assuming $t$ is fixed? And assuming $t$ is part of the input?

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The problem is called Component Order edge-Connectivity. The edge here refers to deletion of edges. See e.g. Gross et al..

The problem is defined as follows. Given an undirected graph $G = (V, E)$ and two integers $k$ and $\ell$, is it possible to delete at most $k$ edges from $G$ such that in the resulting graph, each connected component has at most $\ell$ vertices.

The problem is related to the vulnerability measure Edge Integrity, where you minimize the sum of the largest connected component and the size of the cuts. See e.g. the survey by Bagga et al..

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