$**$SEE ATTACHED IMAGE FOR QUESTION PARTS$**$

For any flow network $G=(V,E),$ we define $G^{+}$as the subgraph of

$G$ with vertex set $V$ and edge set equal to the positive$-$capacity edges in $E.$

For any flow network $G=(V,E)$ and any vertex $v$ in $V,$ we define $P(G,v)$ as the set

of all shortest paths from $v$ to $t$ in $G^1,$ and we define $d(G,v)$ as the length of any path

in $P(G,v)($or $$ if $P(G,v)$ is empty$).$

For any flow network $G=(V,E),$ we define $(G)$ as the set of all edges $(u,v)$ in $G^{!}$

such that $d(G,v)=d(G,u)-1.$

For any flow network $G=(V,E),$ we define $F(G)$ as the set of all flows $f$ in $G$ such

that $f(e)=0$ for all $e$ in $\frac{E}{\frac{?}{?}}(G).$

Fact $1$: For any flow network $G=(V,E),$ any flow $f$ in $F(G),$ and any edge $(u,v)$ in

$(G),$ we have $c_f(u,v)=c(u,v)-f(u,v).$

For any flow network $G,$ we define $F^{**}(G)$ as the set of all flows $f$ in $F(G)$ such that

no path in $P(G,s)$ belongs to $G_f^{\text{I}}.$

For any flow network $G=(V,E),$ any flow $f$ in $F(G),$ and vertex $v$ in $V,$ we define

the predicate $Q(G,f,v)$ as $"$at least one path in $P(G,v)$ appears in $G_f^1".$