Testing grandparents can be important to establish paternity (or maternity) when samples aren’t available from father (or mother) for whatever reason. It can be useful in questions paternity or for inheritance or immigration or just to confirm what are true relationships for peace-of-mind when there are doubts. The approach generally taken is a bit like DNA Testing Siblings using the standard STR approach.
In an ideal world to demonstrate a grandparent grandchild relationship you’d test the grandchild, both of his or her parents as well as both of the grandparents, ie testing all five people in the mini family tree. This will show a genetic link between the child and the parents and also the link between the parents and grandparents. Any statistical evaluation of matching results would be very powerful and non-matches would be 100% conclusive. If, however, there is no sample from father then the information isn’t as complete but matches will still generate strong statistical support. If we lose further individuals from the family tree then the statistical power reduces sharply. If there is only one sample from one grandparent and one parent then results will be limited and might not be very informative. In a situation where there is only a grandchild with a single grandparent then there may not be very much more DNA sharing between them as compared with someone who is no biological relation at all and as a consequence the test will of little or no use at all.
So as with siblings, depending on the size of your purse, it might be advisable to purchase mtDNA and/or Y-STR tests in addition to standard STRs as appropriate to confirm lineages and really improve the level of certainty that can be achieved.