Generalized cover ideals and the persistence property

Let I be a square-free monomial ideal in R=k[x1,…,xn], and consider the sets of associated primes Ass(Is) for all integers s⩾1. Although it is known that the sets of associated primes of powers of I eventually stabilize, there are few results about the power at which this stabilization occurs (known as the index of stability). We introduce a family of square-free monomial ideals that can be associated to a finite simple graph G that generalizes the cover ideal construction. When G is a tree, we explicitly determine Ass(Is) for all s⩾1. As consequences, not only can we compute the index of stability, we can also show that this family of ideals has the persistence property.