she must take a polygraph test.
according to her professor they are not reliable.
even if it says she was lying that the polygraph test could be wrong. How would I respond at that point?
A lot of people have posted here about the accuracy of polygraph. You can go on the internet and find statements about accuracy. From what I can see, the statements are based on "studies." You and her both are smart enough to know that studies are based on circumstances and situations and subjects. The other day there was a study that drinking coffee can help you lose weight. Then there is another study a year ago that says the opposite. Both studies are accurate, but maybe the results are different because of different methods.
There can be bias in a study, too, so you even have to look at who does the "study." For example, about two decades ago there was a study that showed that smoking cigarettes did not show any evidence it causes cancer. The study was performed by a group called something like "The American Health Association for Truth." If you care enough to look and find out, you discover that the "Association" is funded largely by cigarette manufacturers. I'm just making up some of these names, but I'm not making up the problems with "studies." But if you watch a six-paragraph article on the internet, people all believe it if the article says "a recent study says."
Like you said, polygraph is used by the government in some cases. Here is my personal opinion about the polygraph and its use and accuracy and "studies."
I think the studies are based on casual lying. The subjects are asked questions, and then either lie or tell the truth, and then observe if the polygraph is accurate or not. The subject is not under intense scrutiny with a weighty personal investment, such as possibly losing a job, possibly going to jail, possibly losing their marriage. The subject, if they fail the test, know there is no consequence. That alone to me would discount the accuracy of the polygraph. This is my opinion.
Also, when the government uses polygraph, that is just one of several or many methods to obtain the truth and determine the facts. Often they have other evidence, coincidences, interrogations, interviews, eyewitnesses, etc. Contrast this with a "study."
My opinion also - you will never, ever know if the polygraph is accurate or not unless you know the truth independently of the polygraph. You know your wife's name, date of birth, her address, so if she lies and she passes, you know the polygraph didn't work. But if you ask "did you have sex?" and you don't know that answer, you will never know if she passed or not. But you do have one more piece of evidence, one way or the other.
From my observation here, I have not read that anyone ever posted that they regretted the polygraph. I have heard of a couple times here that the polygraph was false, but in those cases the cheater passed the test and later it was discovered they actually cheated. It has been a while so I don't remember the details well.
The main thing found here, I think, is what I would call the "fear detector" effect - the cheater is worried about failing the polygraph, so the cheater comes clean before the polygraph even occurs. It also is called a "parking lot confession" because frequently that is when the cheater finally comes clean - actually in the parking lot about to go into the polygraph examiner's office.
Lastly, just ASKING for a polygraph can give good results. For one, it says it in a certain way, very strongly, that you don't believe her. You can tell her that, "I don't believe you," but telling her "I want you to take a polygraph" takes that to a whole different level. It lets her know how bothered you are, how serious you are, about finding out the truth, and how willing you are to get there. I, personally, would put a voice-activated recorder in her car before you start asking for things you want, because she will call the other man or a confidant and you will find out the truth much better than the polygraph itself.
The other thing about just ASKING about the polygraph, you get to see her reaction. You get to kind of "take her temperature" to see if she really does want to do what is needed to fix the trust issue, and help you through this, be supportive of you, or if she has no sympathy or empathy and just thinks of herself and her feelings. For example, she could say, "I understand why you want to do this, but I have some hesitancy about the accuracy of it" or she could say, "No way I will do that, I'm completely offended by that, how dare you try to do that, you are way off base ... ."
I'm going to tell you flat out that there is a bit more to it than she has told you so far, that there is some level of emotions with this guy, and that she will be in contact with him again in the near future. I think since now that you are aware, you will catch her, or you will at least see how shady she is acting, so you'll know at least if it's still an ongoing affair, regardless of the polygraph.
Go find a thread called "It was my fault" by DazedAndLost and read a bit, especially in the first couple pages and the last couple of pages.
Revelation #3
She had signed up to a "Married Cheaters" style website, where WS's can look for "quickies" in the area. She says she met up with 3 guys from the site. There had been kissing but nothing else.
See where it actually went. This is fairly typical here - the lying and minimizing and half-truths. There are any number of posters here who have read so much they can spot this stuff from a mile away. Your wife has not come fully clean. Pretty sure she will be in touch with him tomorrow or the next day. Much more likely than not she has what I refer to as "you-go girls" encouraging her, or at the least enjoying it vicariously. Very likely there is some stuff that has been going on, at least flirty, going back a while, this didn't just pop up one night, she had thought about it or he had pursued before this.
even if it says she was lying that the polygraph test could be wrong. How would I respond at that point?
I would tell her that "yes, the polygraph is controversial as to whether it is accurate or not to tell if someone is telling the truth or not. Also asking a liar also is not accurate, a person who has lied repeatedly, as you have during this affair, also has a terrible accuracy of telling the truth. So I am using the best possible method I can find, which is NOT YOU, but is the polygraph, as to the truth of the matter."