$($Flow variations$)$ Let G $=\{$V$,$ ~E$,$ ce$,$ wv $,$ s$,$ t$\}$ be a flow network. To each vertex v in V $\backslash \{$s$,$ t$\},$ we associate a

weight, wv $>0.$ We want to compute a flow f of maximum value satisfying the following extra constraint:

$$v in V $\backslash \{$s$,$ t$\}$ the flow entering v must be at most wv $.$

Reduce this variation of the maximum flow problem to an input that can be solved running the Ford$-$Fulkerson

Algorithm. Also, briefly justify why your reduction is indeed an optimal solution to the given problem.

We expect: a detailed explanation of the transformation of the given input to a network that will serve as

input to Ford$-$Fulkerson Algorithm, as well as how you recover a solution to the given problem from the

max flow returned by Ford$-$Fulkerson Algorithm.