Let N be a network which, contrary to our in$-$class definition, has several sources and sinks$.$

Suppose x$1,$ x$2,...,$ xn are the sources and y$1,...,$ ym are the sinks$.$ Define a new network

$$N $=$N $($x$,$ y$)$ in the following way:

$$ Starting from N $,$ add two new vertices x and y$.$

$$ For each i $=1,...,$ n$,$ add a directed edge from x to xi and set its capacity to $.$

$$ For each j $=1,...,$ m$,$ add a directed edge from yj to y and set its capacity to $.$

So $$N is a network with only one source x and one sink y$.$ The capacities above are made

infinite in order to accommodate any possible rate of flow.

Now, suppose f is any flow in N $.$ Extend f to a new flow $$f in $$N by letting $$f $($xxi$)=$ f $+($xi$)$

and $$f $($yj y$)=$ f $($yj $)$ for each i $=1,...,$ n and j $=1,...,$ m$.$ An example is drawn below.

Recall that f $+($v$)$ is the total flow out of the vertex v and f $($v$)$ is the total flow into v$.($To

be clear: for any edge e already in N $,$ the flow rate through e is the same for $$f as it was for

f $.$ That is$,$f $($e$)=$ f $($e$).)$

$($a$)$ Show that $$f is actually a flow in $$N : it satisfies both conservativity and feasibility.