  We show that, for an arbitrary graph, a regular ideal of the associated
Leavitt path algebra is also graded. As a consequence, for a row-finite graph,
we obtain that the quotient of the associated Leavitt path by a regular ideal
is again a Leavitt path algebra and that Condition~(L) is preserved by
quotients by regular ideals. Furthermore, we describe the vertex set of a
regular ideal and make a comparison between the theory of regular ideals in
Leavitt path algebras and in graph C*-algebras.
