Which of the following is an example of a simple sentence?

A simple sentence is a complete thought that consists of a subject and a predicate, conveying an idea without any dependent clauses or additional subject-predicate pairs. In this case, "The dog barked at the mailman" is a simple sentence because it has a clear subject, "the dog," and a predicate, "barked at the mailman," which together form a complete thought.

The other options either do not form complete thoughts or lack the necessary components to be classified as simple sentences. For instance, "While the dog barked" is a dependent clause that cannot stand alone as a complete thought. "The dog barked and ran" is a compound sentence, as it contains two independent clauses joined by the conjunction "and." "The barking dog" is simply a noun phrase and lacks a predicate, making it incomplete.