I have been trying, without success, to get the parent of this match (a possible half cousin) to take a DNA test which I am sure would help me in finding my maternal grandfather, long since dead. Am I correct in saying that each subsequent generation reduces by approximately half the match numbers? If so, the parent of my match would show approximately 488cM across 20 segments, very close to my known half cousin.
Would that be sufficient to confirm that the possible first cousin and I share a grandfather?
DNA won't confirm that you share a grandfather. It may confirm that relationship as being within the range of possibilities, along with several other relationships. Even if the probability of a particular relationship is the most likely, that doesn't mean it necessarily applies to your relationship, because if other relationships weren't possible, their probabilities just wouldn't exist.
IMO you would be better off researching the ancestors of both matches, widening their ancestral lines as much as you can by also researching siblings of direct ancestors, their marriages and children, until you find a possible commonality.
Triangulation may help, but again IME, trying to get enough involved parties interested enough to DNA test for that purpose can be difficult if not impossible much of the time, even if you offer to pay for the tests. They are either just not as interested or involved as you might be, or more so with older generations, suspicious of the whole testing scenario and allowing someone that they don't know access to their results.
Triangulation will also only enhance probabilities. Proof may only be possible through validated paper based research, corroborated by DNA results. Even then, proof may not be possible, and you may have to settle for a most likely explanation derived from the evidence that you have been able to obtain.
One of my higher DNA matches is to a man who has not responded to any contact from me at all, and has a tree on Ancestry consisting of nine people. I spent well over a year researching his ancestry, including buying several birth and marriage certificates looking for clues. The closest I have managed to get to any correlation between us is that his GGM had a child (his GM) out of wedlock by an unknown man, and that one of my GG Uncles had two children out of wedlock with his GGM's sister whilst they were all living in the same street. My GG Uncle also had the same occupation as the apparently fictitious father named on my match's GM's birth certificate.
Proof that we are related through my GG Uncle? No, but on the balance of the evidence available, it seems to be the most likely explanation. It could have been another of my GG Uncles, there were several in the area at the time, but opportunity and the occupation on the birth certificate lead me to that specific GG Uncle as the likely candidate.
Our match length is quite high for that scenario, but the research has not uncovered evidence for any closer relationship, and no likelihood of our families paths crossing a second and more recent time, other than by completely random chance and no opportunity amongst my closer ancestors, apart from perhaps one great uncle who seems to have disappeared off the face of the earth from the age of 14.
In the end though, I doubt that any cooperation from my DNA match would have resolved that any better, except perhaps to have saved my quite a bit of time and expense along the way.
Have you checked the shared matches between you and your matches, if there are any? Researching their lines and connections could also lead to commonalities that may point in the direction of, and further corroborate a possible identity for you GF.
It's all a lot of hard work, time and effort unfortunately. There are no easy short cuts, but of course if you do eventually find an answer, it makes everything worthwhile.
