Here is the fragment again:
The neighborhood cat whom even the biggest dogs fear.
You correctly identified the fragment as an appositive phrase. Cat, the noun that begins the phrase, is the clue.
To fix this fragment, you could add a verb and finish the thought, like this:
The neighborhood cat whom even the biggest dogs fear
surveyed the street that he ruled.
Or you could use Punctuation Rule 9 to fix the problem. With an appositive, no matter where you put it, you need to use commas, like this:
Rocky,
the neighborhood cat whom even the biggest dogs fear,
surveyed the street that he ruled.
Or even this:
Mrs. Paul let out Rocky,
the neighborhood cat whom even the biggest dogs fear.
Enjoy your wealth!