$(50)[$Flow network: quiz$]$ Modified from Textbook Exercise $5$ in Chapter $7.$ The statement

in the Exercise $5$ is false, and a counterexample flow network is given below.

Write your answer as specified below.

a$)$ Run the basic Ford$-$Fulkerson max flow algorithm on the counterexample flow network

above, and show the resulting flow network with flow$/$capacity $($e$.$g$.,1/1)$ for each edge and

show the resulting residual graph with applicable $($i$.$e$.,>0)$ forward and backward residual

capacities for each edge. State the maximum flow value and the minimum cut $($e$.$g$.,(,$

c$\},\{d,1\}.$

b$)$ Draw the flow network above modified by adding $1$ to every capacity. Then, run the basic

Ford$-$Fulkerson max flow algorithm on the modified flow network, and show the resulting

flow network with flow$/$capacity for each edge and show the resulting residual graph with

applicable forward and backward residual capacities for each edge. State the maximum flow

value and the minimum cut. Note this minimum cut is different from the minimum cut from

the step a$.$