Which of the following sentences correctly uses the genitive noun case?

The sentence "It is the dog's toy." correctly uses the genitive noun case, which is utilized to show possession or belonging. In this context, the genitive is indicated by the structure "the dog's," where the apostrophe followed by "s" signifies that the toy belongs to the dog. This construction effectively conveys the relationship of ownership.

The other sentences do not demonstrate the genitive case as they either describe actions without denoting possession or are simply declarative statements. For example, the first sentence discusses the dogs barking, the third indicates where they played, and the fourth describes someone's running speed. These do not involve a possessive form; hence they cannot be classified under the genitive noun case.