A digraph $\backslash ($ G $\backslash )$ is said to be monopathic if for every pair of its distinct vertices $\backslash ($ u $\backslash )$ and $\backslash ($ v $\backslash )$ there is at most one simple path directed from $\backslash (\backslash$boldsymbol$\{$u$\}\backslash )$ to $\backslash ($ v $\backslash )($and at most one from $\backslash (\backslash$boldsymbol$\{$v$\}\backslash )$ to $\backslash (\backslash$boldsymbol$\{$u$\}\backslash )).$

Now let $\backslash ($ G $\backslash )$ be an arbitrary strongly connected digraph. How many of the following five statements are true?

$1.\backslash ($ G $\backslash )$ is monopathic if and only if $\backslash ($ G$^\{$T$\}\backslash )$ is monopathic.

$2.\backslash ($ G $\backslash )$ is monopathic if and only if both $\backslash (\backslash$operatorname$\{$DFS$\}($G$)\backslash )$ and $\backslash (\backslash$operatorname$\{$DFS$\}\backslash$left$($G$^\{$T$\}\backslash$right$)\backslash )$ have no forward$-$edges and no cross$-$edges.

$3.\backslash ($ G $\backslash )$ is monopathic if and only if any pair of distinct simple cycles in $\backslash ($ G $\backslash )$ have at most one vertex in common.

$4.\backslash ($ G $\backslash )$ is monopathic if and only if removal of any single non$-$self$-$loop edge from $\backslash ($ G $\backslash )$ will make it not strongly connected.

$5.\backslash ($ G $\backslash )$ is monopathic if and only if $\backslash ($ D F S$($G$)\backslash )$ has no forward$-$edges and no cross$-$edges, and for each vertex $\backslash ($ u $\backslash )$ other than the DFS$-$root, there is a unique back$-$edge $(\backslash ($ x$,$ y $\backslash ))$ such that $\backslash (\backslash$boldsymbol$\{$x$\}\backslash )$ is a descendant of $\backslash (\backslash$boldsymbol$\{$u$\}\backslash )$ and $\backslash (\backslash$boldsymbol$\{$y$\}\backslash )$ is a proper ancestor of $\backslash (\backslash$boldsymbol$\{$u$\}\backslash )($with respect to the DFS$-$tree$).$a$.3.$b$.0.$C$.4.$d$.2.$e$.1.$f$.5.$