Questions

$(20$ points$)$: Consider a graph $G$ consisting of vertices $V=\{v_1,v_2,v_3,v_4,v_5\}$ and edges $E=\{\begin{array}{c}{e}_1=\end{array}\{$:$(v_1,v_2),e_2=(v_2,v_3),e_3=(v_3,v_1),e_4=(v_2,v_4),e_5=(v_4,v_5),e_6=(v_4,v_4)\}.$

$($a$)$ Assume $G$ is directed. Are there any circuits in $G?$ If so$,$ identify them.

$($b$)$ Draw $G$ and label each vertex and edge accordingly.

$($c$)$ Determine the in$-$degree and out$-$degree of each vertex in $G.$

$($d$)$ Create an induced sub$-$graph $F$ by removing $v_2$

$($e$)$ Has $F$ different connected components $?$