The second sentence is an independent clause!
Answer:
Running after her dog, she lost a shoe.
Explanation:
A participial phrase consists of a participle, modifiers and complements.
Running after her dog is a participial phrase consisting of the participle running and the prepositional phrase <em>after her dog, </em>which includes the preposition <em>after,</em> the object of the preposition <em>dog</em>, and the possessive adjective <em>her</em>.
