Consider the following collection of edges from the network in $3$Consider the following graph, which describes a flow network.

Here, the vertex labeled $So$ is the source, and the vertex labeled $Si$ is the sink. The

edge weightings indicate flow capacity, and the figures in brackets indicate the current

flow through the network.

$C=\{(So,v_1),(v_3,v_8),(v_5,v_7)\}.$

a$)$ Explain briefly why this set of edges forms a cut for the network. $[4$ marks$]$

b$)$ What is the capacity of this cut?

$[1$ mark$]$

c$)$ Does there exist a cut for the network with a lower capacity than $C?$ If so$,$

find the cut with minimum capacity. If not, explain why such a lower capacity

cut cannot exist.

$[5$ marks$]($ii$)$: C $=\{($v$1,$ v$6),($v$2,$ v$4),($v$2,$ v$5),($v$3,$ v$8)\}.$ a$)$ Explain why C is a cut for the network. What is the capacity of C$?[4$ marks$]$ b$)$ Decide, with justification, whether a cut with smaller capacity exists.