2
$\begingroup$

If we do not allow unit clauses, can 2CNF containing 3 occurences per variable be unsatisfiable?

Every time I try to connect opposing variables to a cycle, I need to draw 4 edges for at least one variable (2 per positive and negative occurences).

So, is it even possible?

Improve this question
$\endgroup$
1
  • $\begingroup$ Are you allowing less than 3 occurrences? $\endgroup$
    – Yuval Filmus
    Sep 1, 2017 at 8:18

1 Answer 1

Reset to default
2
$\begingroup$

The following is unsatisfiable: $$ (x \lor y) \land (x \lor \lnot y) \land (\lnot x \lor z) \land (\lnot z \lor w) \land (\lnot z \lor \lnot w) \land (y \lor w). $$ This contains every variable exactly three times, not all of them of the same polarity. If you allow variables to appear less than three times, you can drop the last clause $y \lor w$.

Improve this answer
$\endgroup$
1
  • $\begingroup$ How could I miss that simple example with 2 dummy variables... $\endgroup$
    – rus9384
    Sep 1, 2017 at 9:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.