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I have a connected graph with $nk$ vertices, and would like to select $n$ disjoint induced subgraphs with $k$ vertices such that each subgraph is connected, selecting one out of all possible solutions at random. It doesn't necessarily need to be uniformly at random, I just want the possibility of getting any possible solution.

For example, with this graph:

a cycle of 4 vertices, sharing 1 vertex with a cycle of 3 vertices

and $k=3$, $n=2$, here is a solution (in fact it's the only one):

the vertices in the 3-cycle are grouped together, and so are the remaining 3 vertices

And here's one of the two possible solutions for the same graph with $k = 2$ and $n = 3$:

two pairs of vertices in the 4-cycle are grouped together, as are the remaining 2 vertices

Sometimes this is not possible, like with this graph, for $k = 3$ and $n = 2$:

a path of 4 vertices, with a path of 2 vertices coming out of the second vertex

Is there any (efficient) way of determining whether there is a solution, and if so, selecting one (not necessarily uniformly) at random?

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    $\begingroup$ Checking whether the partition exists is NP-hard: "On the complexity of partitioning graphs into connected subgraphs" has some references and shows that this holds even for planar graphs. $\endgroup$
    – user114966
    Apr 18 at 18:43
  • $\begingroup$ @Dmitry Thanks! I figured this had already been looked into before, but wasn't sure what to google... $\endgroup$
    – pommicket
    Apr 18 at 18:53
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The problem is unfortunately NP-hard. Thanks to Dmitry for finding this: https://www.sciencedirect.com/science/article/pii/0166218X85900083.

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