Consider an edge$-$weighted graph G in which edge weights are all distinct from each

other. Let T be a MST of G$.$ Suppose the weight of one edge e$0$ in E$($G$)$ is changed fromw$($e$0)$ to w$0($e$0)$ in which w$0$

$($e$0)$ is a unique weight different from w$($e$)$ for any e in E$($G$).$ The weights of other edges in E$($G$)$ remain same. So a new graph G$0$ and a new weight function w$0$ are defined such that $($i$)$ V $($G$0)=$ V $($G$),$

$($ii$)$ E$($G$0)=$ E$($G$),($iii$)$ w$0$

$($e$)=$ w$($e$)$ for each e $6=$ e$0$ in E$($G$),$ and $($iv$)$ w$0($e$0)6=$ w$($e$)$ for any e in E$($G$).$

We are interested in deciding whether e$0$ in T$0$ where T$0$ is a MST of G$0.$ Answer true or false

in each of the following cases. You do not need to prove your answer.

$($a$)$ e$0$ is always in T$0$ when e$0$ in T and w$0($e$0)<$ w$($e$0)$

$($b$)$ e$0$ is always in T$0$ when e$0$ in T and w$0($e$0)>$ w$($e$0)$

$($c$)$ e$0$ is always in T$0$ when e$0$ in $/$ T and w$0($e$0)<$ w$($e$0)$

$($d$)$ e$0$ is always in T$0$ when e$0$ in $/$ T and w$0($e$0)>$ w$($e$0)$