Suppose you are given a directed graph G $=($V$,$ E$)$ with V $=\{1,2,3,4,5,6,7,8\}$

and the depth first intervals $([$pre$,$ post$])$ of each vertex are as follows. $\{1$ : $[4,5],$

$2$ : $[7,8],3$ : $[12,13],4$ :$[14,15],5$ : $[3,6],6$ : $[2,9],7$ : $[1,10],8$ : $[11,16]\}.$

$($a$)$ Draw this directed graph using networkx package in Python and include the image.

$($b$)$ What are the descendent and ancestor vertices of vertex $6?$

$($c$)$ How many connected components does the graph have?

$($d$)$Identify three pairs of vertices that form a cross edge $($i$.$e$.,$ one is neither a descendent nor

an ancestor of the other$).$