  If $E$ is a graph and $K$ is a field, we consider an ideal $I$ of the Leavitt
path algebra $L_K(E)$ of $E$ over $K$. We describe the admissible pair
corresponding to the smallest graded ideal which contains $I$ where the grading
in question is the natural grading of $L_K(E)$ by $\mathbb Z$. Using this
description, we show that the right and the left annihilators of $I$ are equal
(which can be somewhat surprising given that $I$ may not be self-adjoint). In
particular, we establish that both annihilators correspond to the same
admissible pair and its description produces the characterization from the
title. Then, we turn to the property that the right (equivalently left)
annihilator of any ideal is a direct summand and recall that a unital ring with
this property is said to be quasi-Baer. We exhibit a condition on $E$ which is
equivalent to unital $L_K(E)$ having this property.
