The problem is they are in the same Order, but different Genus, it would be like testing human DNA to a Bonobo or Chimpanzee there would be very little to tell someone who was who.
Sex chromosomes are easy kids do that in high school, DNA mapping is complex.
Now how many chromosomes they have might tell a tale, like Horses vs Donkeys vs mules...
Most hybrids (in the same Genus, different genus is still not known to happen in the upper animal kingdom, (reptiles, mammals, birds)) have an odd number of chromosomes because the parents have different numbers of chromosomes, and one parent passes on one that doesn't pair up...
So that would be a way to tell- you would need blood or... really skin or blood for the best results.
ALL geese -should- have the same number, all swans, all ducks, each group would have its own number so if the two groups mixed the russulting offspring would have the 'average' of the two parents (rounded up to the nearest whole number).
(EDIT: tail/tale, *shrug*)