  For any field $K$ and for a completely arbitrary graph $E$, we characterize
the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of
two-sided ideals) in terms of the underlying graph. When the algebra
decomposes, it actually does so as a direct sum of Leavitt path algebras for
some suitable graphs. Under certain finiteness conditions, a unique
indecomposable decomposition exists.
