Sentence fragment

I have a question about fragment sentences, that is sentences that lack verb or subject. Let's take those two examples from Manhattan sentence correction strategy guide:

1. The cat sitting by the stairs watched the mouse.
2. A new textbook focused on recent advances in artificial intelligence assigned by our instructor.

According to Manhattan's explanation, the first sentence is grammatically correct as it includes a subject (cat) and a verb (watched). On the contrary, the second sentence is fragment because it lacks a main verb. Up to their explanation, for the sentence to be grammatically correct , the sentence should written as "A new textbook ... was assigned" (we should add "was" before the word assigned).

I don't understand what is the difference - in structure - between the two sentences that make's the first sentence right and the second sentence wrong. In other words, why should we add the word "was" to the second question but not to the first sentence?

Thanks in advance!