Figure $7\mathrm{.}24$ What are the

minimum $s-t$ cuts in this flow

network?

Figure $7\mathrm{.}25$ What is the min$-$

imum capacity of an $s-t$ cut in

this flow network?

$(50)[$Minimum cut$]$ Do the textbook exercise $1$ in Chapter $7$ as instructed below.

a$)$ For the exercise a$,$ there are multiple minimum s$-$t cuts.

$($i$)$ List all minimum cuts found through examination

$($ii$)$ Show the minimum cut found by the Ford$-$Fulkerson algorithm; additionally, show the

initial flow network $($G$)$ and residual graph $($Gf$)$ and the final flow network $($G$)$ and

residual graph $($Gf$).$ Note the Ford$-$Fulkerson algorithm finds one of the minimum cuts.

b$)$ For the exercise b$,$ there is a unique minimum s$-$t cut. Show the minimum cut and its capacity

found using the Ford$-$Fulkerson algorithm; additionally, show the initial flow network $($G$)$

and residual graph $($Gf$)$ and the final flow network $($G$)$ and the residual graph $($Gf$).$

In your answer, show a minimum cut as a pair $($A$,$ B$),$ for example, $(\{$a$,$ b$,$ c$\},\{$d$,$ e$\})$ where $\{$a$,$b$,$ c$\}$ is a set of vertices on the source side and $\{$d$,$ e$\}$ is the set of vertices on the sink side.