Trace through the Ford$-$Fulkerson max flow algorithm on the input instance given in the figure below. Note that in the first iteration when value of the flow is $0,$ the graph G and the residual graph Gy are one in the same. Start by highlighting the first augmenting path in the figure below, labeled Iteration $1.$ For iterations $2$ and later, indicate $($using the boxes$)$ the current$/$new flow f in G and give the value of the low v$($f$).$ Then $($in the provided space labeled G$,)$ draw the residual graph for that flow, and highlight in G$,$ the next augmenting s$-$t path, giving the bottleneck edge capacity b of the augmenting path $($i$.$e$.,$ the amount by which you are augmenting the flow value$).$ For full credit, always greedily choose the greatest$-$possible augmenting path from G$,$ in each iteration. $($Le$.,$ if there are two augmenting paths, one with bottleneck value $5,$ and the other with bottleneck value $8,$ you should choose the latter.$)$ Again, you should start with simply highlighting the first augmenting s $-$ t path in the top figure for Iteration $1$ here, and giving its bottleneck capacity. Graphs for iterations $3$ and onward are provided on the next page.

$($b$)$ Certify that the flow you illustrated in part $($a$)$ is indeed the maximum flow, by finding a min s $-$t cut with capacity equal to the flow value. Indicate the cut both on the graph as well as by writing down the partition of the nodes in the space below. $($c$)$ You are working for a company that manages a large flow network that coincidentally looks just like the one in part $($a$).$ Business is excellent, and this maximum flow is insufficient to carry all the flow they have orders for. The company proposes to select an edge e$\text{'}$ and increase the capacity of this edge by $1$ unit in order to increase the value of the maximum flow. You warn the company to be careful which edge is chosen for this increased capacity, as not all edges will increase the value of the maximum flow.

$($i$)$ Prove your point, by giving a choice of e$\text{'}$ in the graph of part $($a$)$ that will not increase the maximum flow value. Le$.,$ after the capacity of e$\text{'}$ is increased by $1,$ the max flow is the same as before the capacity of the edge e$\text{'}$ was increased. $($ii$)$ Now give a choice of e$\text{'}$ in the graph of part $($a$)$ that would increase the maximum flow value.