Nah she doesnt. Edward described it several times, her power [which is nothing supernatural] is her প্রণয় for the family. Plus if আপনি check the vampire index on the back of Breaking Dawn, আপনি will see she is not marked
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I don't think it's solely reserved for her family though. I think it's just her ability to প্রণয় anyone passionately like Edward says on Twilight page 307.