Given non-simple graph ( G = (V,E) ), if we construct the augmented simple graph ( G’ ) by adding two nodes on every edge, then ( G’ ) has a minimum cut of size ( 2|E| + k ) if and only if ( G ) has a minimum cut of size ( k ).

I understand the new augmented graph has twice as many edges and nodes.

Additionally, each edge is replaced by three edges, and I comprehend that at least two of these new edges can be part of the cut.

However, I’m struggling to find a way to connect it to a correct solution.
I would be grateful if you could help me with this problem.