Fragment or Sentence

Question

A course book presents the bolded part of the below sentence pairs as examples of fragments:

• I am planning to go for a new smart phone. The one I am using is dead slow.

• An MNC has expressed interest in recruiting our students for back office positions. Their HR managers are expected to visit our college soon.

But my instinct is that both of them are full sentences, not fragments. They do not come within the definition of a fragment, i. e. "A SENTENCE FRAGMENT fails to be a sentence in the sense that it cannot stand by itself. It does not contain even one  independent clause". 

Edit:

In grammar anaphora is the use of a word referring back to a word used earlier in a text or conversation, to avoid repetition. In both the "sentences/fragments", there are anaphoric pronouns.

Can the anaphoric pronouns make sentences fragments?

Are they actually sentence fragments?

Answer 1

Grammatically, they are not sentence fragments. They have a subject and predicate. Content-wise, they are incomplete in that they have pronouns that refer to antecedents from other sentences (in the first example, you can't know that "the one" refers to a phone reading just the bolded sentence; in the second example, reading just the bolded sentence isn't enough to know that "their" refers to the MNC), but that doesn't make them fragments in the grammatical sense.