There are a few misconceptions that some other commenters are already making.

This min-cut is different from the max-flow min-cut theorem's min-cut. This is what referred to as global min-cut, which does not fix the terminals. In max-flow min-cut theorem, it is about disconnecting two fixed vertices s and t. In the global min-cut, it is about disconnecting two arbitrary vertices.

Source: thesis was on hypergraph min-cut

Can someone give a more concrete example of this problem?

I read the wiki page[1], and I'm left with the impression that for almost all cases, the minimum cut will be to pick a node and seperate it from the rest of the nodes. It doesn't sound like a very computationally difficult or useful algorithm if one of the two parts of the results usually consists of a single node...

[1]: https://en.wikipedia.org/wiki/Minimum_cut

On that note, here is the currently fastest implementation for the problem https://arxiv.org/abs/1808.05458 and here on github https://github.com/viecut/viecut

Question: is this algorithm practical to implement?

What is it used for

I remember a CS prof saying he moved from hardware to software because the speedups possible in algorithm design dwarf anything you can achieve by improving the bare metal.

[2019]