I would refuse to take the polygraph, and if that prevents you from having upward movement in the company I would get a lawyer. A Polygraph test is NOT a scientifically proven test. Basically, it relies on your physical reaction to certain questions. If you are lying and you have forced yourself to believe it to be true, your body will react as if it was true and you will easily pass the lie. On the opposite end, if you are generally nervous when asked questions whether it's true or not, the polygraph would probably state that you are lying.
If anybody was ever fired or lost their job because of a polygraph test alone, they would have a huge lawsuit to file. In court, polygraphs are considered incredibly circumstantial evidence. If you have a bunch of witnesses, DNA evidence, and a polygraph, then combined it would be a good case against the defendant. On its own, a polygraph result is lousy evidence because they are VERY commonly tricked.