Problem $2(28$ points$).$ Please answer the following:

$($i$)$ Bob says that for each positive integer $\backslash ($ k $\backslash ),$ every graph with more than $\backslash ($ k$^\{2\}\backslash )$ edges fails to have a vertex cover of size less than or equal to $\backslash ($ k $\backslash ).$ Is he correct?

$($ii$)$ Given a graph $\backslash ($ G$=($V$,$ E$)\backslash )$ and a positive integer $\backslash ($ k $\backslash ),$ the algorithm below determines whether $\backslash ($ G $\backslash )$ has a vertex cover of size less than or equal to $\backslash ($ k $\backslash ).$

$11$: $\backslash (\backslash$quad $\backslash )$ Go to line $14$;

$12$: end if

$13$: end while

$14$: Determine by brute force whether $\backslash ($ G $\backslash )$ has a vertex cover of size less than or equal to $\backslash ($ k $\backslash )$;

Please explain why it is correct to return $"$no$"$ in line $9.$