Ideals in Graph Algebras

We show that the graph construction used to prove that a gauge-invariant ideal of a graph C*-algebra is isomorphic to a graph C*-algebra, and also used to prove that a graded ideal of a Leavitt path algebra is isomorphic to a Leavitt path algebra, is incorrect as stated in the literature. We give a new graph construction to remedy this problem, and prove that it can be used to realize a gauge-invariant ideal (respectively, a graded ideal) as a graph C*-algebra (respectively, a Leavitt path algebra).
Comment: Version II Comments: The graphs displayed in Example 3.2 and Example 4.2 were incorrect and are now fixed. Some small typos corrected. Version III Comments: Some small typos corrected and additional references added. Version IV Comments: Minor changes to exposition, some typos corrected, some references added. To appear in Algebras and Representation Theory