Problem #WSP-000250

Problem

Noah has \(10\) dogs, who he wishes to group into \(5\) pairs for \(5\) families, each of whom want two dogs. However, the dogs are quite picky, and can’t be paired with most of the other dogs. None in the first group will go with each other: an alsatian, a border collie, a chihuahua, a dachshund and an English bulldog. None in the second group will go with each other: a foxhound, a greyhound, a harrier, an Irish setter and a Jack Russell. Furthermore,
The alsatian is the least picky and can be paired with any in the second group.

The border collie won’t go with the foxhound, but will go with any other dog in the second group.

The chihuahua and the dachshund will only go with the Irish setter and the Jack Russell.

Additionally, none of the foxhound, greyhound and harrier will go with the English bulldog.
Is it possible to pair up the \(10\) dogs?