Polynomial algo for 2-SAT related algorithm

Question

I read many algorithm for finding the 2-SAT problem, i.e. given expression is satisfiable or not, which can be solved in polynomial time. example(algorithm).
For Krom's procedure(Wikipedia),I construct graph from which I can easily verify its satisfiability in polynomial time.
Now, I want to find solution of this expression to be satisfiable.
I am thinking like that (verify it): first I take any literal of expression that forms strong connected component say x,and assign value as 0. Then according to algo, there exist edge ( x! -> y). therefore y cannot be 0. (since 1 -> 0 is false) and similarly proceeding if possible.
Otherwise take x=0 and then have only choice as y=1 and similarly proceed to get any 1 instance for which it is satisfiable.

Now, Is any feasible solution of finding any one of this in polynomial time

• give all possible solutions for which expression is satisfiable.
• Or is this expression is satisfiable for input having total k literals = 1.
• Or how many minimum number of literals have value 1 for expression to be satisfiable.

I am not getting any polynomial algorithm for this types of questions.

Answer 1

give all possible solutions for which expression is satisfiable

No, because there can be exponentially many.

Or is this expression is satisfiable for input having total k literals = 1

No, because if this were easy, then so too would be weighted 2-satisfiability (NP-hard).

Or how many minimum number of literals have value 1 for expression to be satisfiable

This is weighted 2-SAT.