Let us assume that we have a 2CNF $\varphi(X,y)$. Then we want to see if there is equal formula where $y$ (or $\overline y$) is pure literal. Can this be done in polynomial time? Are there some criteria for understanding this? Can we say if any minimal CNF has $y$ as pure literal, then all of them do?
$\varphi(x_1,x_2,y)=(x_1\lor y)\land(x_1\lor\overline y)\land(x_2\lor y)$.
Equal minimal 2CNF: $x_1\land(x_2\lor y)$.
Pure literal - literal that has only negative/positive appearances in formula.
Equal formula. We say formulas $\varphi$ and $\psi$ are equal if $\varphi\oplus\psi\equiv0$.
Minimal formula - shortest equal formula.
P.S. Last question is the most significant because it also answers if minimal 2CNF has all pure literals.