"The children's dogs"
We never use the definite after nouns in the genitive, just like you don't in English. You wouldn't say the children's the dogs. It's the same in Swedish.
It's also the same with possessive pronouns. We say min bok 'my book' and you cannot have a definite noun instead in either language – you can't say "my the book" and you couldn't have boken after min in Swedish either.
It could be explained along these lines: the basic meaning of the definite form is to say 'you know which one I mean'. But since when you reach the word in question, you have already said something that makes it clear that it is something known, it doesn't make sense to add that grammatical marker. (if this doesn't make sense to you, just disregard it – it's just a suggestion for how to think of it).
"Barnet" is the definitive form of "Barn" (child), so "Barnet" means "The child", because of the "et" at the end. And "Barnen" is the definitive form of "The children". Each time you see a "S" at the end of a noun, it's always going to be a possessive form (like putting a ('s) in english. So "Barnens" means "The Children's"-->somethingpossessed. So "Barnens Hundar" means "The children's dogs".
new question: there is a phrase that is quite the same in my exercise, "Our children´s books", which translates into "våra barns böcker". I guess you will say this one (the children´s dogs) is a definite form, and the other one is not. But i can not understand why the other is not. In italian, we translate the two into: - "i cani dei bambini" (barnens hundar, hund = cane and barn = bambino) - "i nostri libri" which is equivalent to "i libri di noi" (våra = nostri/di noi and böcker = libri) both cases have the same "genitive" ("dei bambini" = "di noi"), so it is quite hard to get why I can not put "våra barnens böcker"