# How to check wether a 2SAT implication graph has a bad loop or not?

m doing a practise quiz which my professor has posted online, one of the questions im very confused about was this one.

   Consider the clauses {x,y} , {~x,y} , {~y,z} , {~z,w} , {~w,~y}
is this true or false?

The implication graph contains the following bad loop:
x -> y -> z -> w -> ~y -> x


I said that this statement was false , however it came out that its true, can someone explain to me why this is the case?

Because i thought a bad loop would be a transition between a letter to the negation of its own then back to the original for example x -> ~x -> x

• In this loop you have a path $x\rightsquigarrow y \rightsquigarrow \bar{y} \rightsquigarrow x$. So if you start from $y$, you can reach $\bar{y}$ and return to $y$, by following the path $y \rightsquigarrow \bar{y} \rightsquigarrow x \rightsquigarrow y$. – Ariel Dec 13 '15 at 19:54
• How does that prove the implication graph containing x -> y -> z -> w -> ~y -> x has a bad loop? – Mr_Shoryuken Dec 13 '15 at 20:02