[sltkr]

Also I don't think the equivalence between edge/vertex versions is trivial at all (though maybe we just have different standards of triviality).

For example, in a grid like this:

    ..####
    .....#
    #.#..#
    #...H#
    ######

A single wall placed (i.e. vertex removed) can block two edges, and it's not obvious what graph transformation can turn that into a single edge.

[emil-lp]

You transform it into the directed case and then you turn each vertex into an arc.

There is a standard construction for going between vertex and edge cuts.