Suppose we are given a non$-$negative weighted directed graph G$($V$,$E$)$ and a cost

function c$($e$)$ that computes weight for edge e in E$.$ Let P be a minimum$-$cost s$-$t

path for G$.$ Let c$($e$)$ be a modified function for computing the weights of the

edges. Will P must still be a minimum$-$cost s$-$t path if: $($i$)$ c$($e$)=$ k $*$ c$($e$),$ for

some k$>0,($ii$)$ c$($e$)=$ c$($e$)+$ k$,$ for some k$>0,($iii$)$ c$($e$)=$ c$($e$)-$ k$,$ for some k$>0,$

$($iv$)$ c$($e$)=1/$c$($e$).$ Justify your answer in each case.