$(20$ points$)$ Let $a,b,c,d,e$ be Boolean variables. Consider the logical formula

$=(av\frac{v}{b}ar(b)vvc{)}^{{?}^{?}}(avvbvvd{)}^{{?}^{?}}(vvbvvd{)}^{{?}^{?}}(bvvcv\frac{v}{b}ar(e){)}^{{?}^{?}}(\frac{?}{\text{b}\text{a}\text{r}}(b)v\frac{v}{b}ar(d)vve)$

$($a$)$ Construct the contact network graph that corresponds to the logical formula $.$

$($b$)$ Draw the contact networks representing the truth value assignments $10011$ and $01101.$ Is $$ satisfiable?

Explain your answer.

$($c$)$ Draw a covering graph for $01101.$ Should covering graphs be rejected from the contact network algorithm?

Explain why or why not.

$($d$)$ Using the second molecular algorithm for solving SAT, construct the graph $G()$ that corresponds to the

formula $$ and explain why covering graphs should be rejected by this algorithm.