Every cut is the union of (edge-disjoint) minimal cuts

20 Views Asked by At

I am tasked with proving the following statement:

"Every cut is the union of edge-disjoint minimal cuts"

The only information given, is the existance of the cut-set subspace $W_S(G)$. It seems I have to say something about the fact that $W_S(G)$ is a subspace, thus closed under addition, and then proceed to show that the sum of minimal cut vectors is again a cut.

However, I cannot seem to get how this statement would be true, given a graph with a bridge. In that case, there is only one minimal cut (removing the bridge), but there are numerous other cuts that are clearly not unions of this one minimal cut.

Am I seeing it wrong?