Minimum Spanning Tree

$1.$ For each of the following statements decides whether they are correct. If it is incorrect, provide a counter$-$example. Assume G $=($V$,$ E$)$ is undirected and connected. Do not assume the edge weights are distinct unless specifically stated.

$($a$)$ let e be any edge of minimum weight in G$.$ Then e must be part of some MST$.$

$($b$)$ If G has a cycle with a unique lightest edge e$,$ then e must be part of every MST$.$

$($c$)$ For any r $>0,$ define an r$-$path as a path whose edges all have a weight less than r$.$ If G contains an r$-$path from s to t$,$ then every MST of G must also contain an r$-$path from s to t$.$

Minimum Spanning Tree

$1.$ For each of the following statements decides whether they are correct. If it is incorrect, provide a counter$-$example. Assume G $=($V$,$ E$)$ is undirected and connected. Do not assume the edge weights are distinct unless specifically stated.

$($a$)$ let e be any edge of minimum weight in G$.$ Then e must be part of some MST$.$

$($b$)$ If G has a cycle with a unique lightest edge e$,$ then e must be part of every MST$.$

$($c$)$ For any r $>0,$ define an r$-$path as a path whose edges all have a weight less than r$.$ If G contains an r$-$path from s to t$,$ then every MST of G must also contain an r$-$path from s to t$.$

$2.$ Consider G $=($V$,$ E$)$ is undirected and connected. Prove that If e is part of some MST of G$,$ then it must be the lightest edge across some cut of G