  We introduce the notion of dual perfect bases and dual perfect graphs. We
show that every integrable highest weight module $V_q(\lambda)$ over a quantum
generalized Kac-Moody algebra $U_{q}(\mathcal{g})$ has a dual perfect basis and
its dual perfect graph is isomorphic to the crystal $B(\lambda)$. We also show
that the negative half $U_{q}^{-}(\mathcal{g})$ has a dual perfect basis whose
dual perfect graph is isomorphic to the crystal $B(\infty)$. More generally, we
prove that all the dual perfect graphs of a given dual perfect space are
isomorphic as abstract crystals. Finally, we show that the isomorphism classes
of finitely generated graded projective indecomposable modules over a
Khovanov-Lauda-Rouquier algebra and its cyclotomic quotients form dual perfect
bases for their Grothendieck groups.
