We characterize, in terms of its idempotents, the Leavitt path algebras of an arbitrary graph that satisfies Condition (L) or Condition (NE). In the latter case, we also provide the structure of such algebras. Dual graph techniques are considered and demonstrated to be useful in the approach of the study of Leavitt path algebras of arbitrary graphs. A refining of the so-called Reduction Theorem is achieved and is used to prove that I(Pc(E)), the ideal of the vertices which are base of cycles without exits of the graph E, a construction with a clear parallelism to the socle, is a ring isomorphism invariant for arbitrary Leavitt path algebras. We also determine its structure in any case.
