[sltkr]

That conclusion may be too hasty. If min cut with k vertices is NP-hard on arbitrary graphs, that doesn't automatically mean that that applies to a 2D grid too.

Is NP hardness proven for just planar graphs? Those are closer to the 2D grid, but still slightly more general. All I could find was a reduction to densest k subgraphs, but Wikipedia tells me that whether that problem is NP hard for planar graphs is an open question.

To be clear, I would be very surprised if the problem turns out to be _not_ NP hard, but there is no trivial equivalence to min cut in general graphs to show that it is.

[emil-lp]

I agree, that is a good point. Although it is (induced) subgraphs of 2D grids, which gets you a bit closer to the planar case (albeit with bounded degree).

It might be polytime on planar graphs, but that would be surprising.