$(20$ marks$)$ Explain how to transform the problem of computing a minimum cost vertex cut into an equivalent minimum cut problem. $($So given a flow network G $=($V$,$ E$)$ you must show how to construct another flow network G$=($V $,$ E$)$ so that G has a vertex cut of cost d if and only if G$$ has a minimum cut of capacity d$).$ Assume that G does

not have an edge $($s$,$ t$)$ connecting directly the source and the sink. Show your network for G$$ for the above flow network.

$(16$ marks$)$ Prove that your transformation is correct. $($First state what needs to be proven as a double implication $).$