The Kirchberg-Phillips theorem says that unital separable nuclear purely
infinite simple $C^*$-algebras in the UCT class are classified by their
(topological, $C^*$-algebraic) K-theory and, more generally, that any two
separable nuclear purely infinite simple $C^*$-algebras that are
KK-isomorphic are in fact isomorphic. This result was preceded by a
similar result for Cuntz-Krieger $C^*$-algebras (all of which are separable
and nuclear) due to Cuntz and Rørdam. It is a long-standing open problem
to determine whether a similar result holds for some sizable class of
plain purely infinite simple algebras, including those appearing as
Leavitt path algebras. The Kirchberg-Phillips theorem has two main
ingredients: Kirchberg's Geneva theorems concerning tensor products with
the Cuntz algebras $O_2$ and $O_\infty$, both of which are known to fail in
the purely algebraic case, and the Phillips theorem that describes
KK-groups between any two $C^*$-algebras in the KP class as the homotopy
classes of homomorphisms, and also as asymptotic unitary equivalence
classes of asymptotic homomorphisms. In this talk, I will discuss recent
results which provide a description of algebraic and hermitian bivariant
K-theory between purely infinite simple Leavitt path algebras in terms
of homotopy classes and in terms of generalized conjugacy classes of
homomorphisms, which we use to show that the Bowen-Franks group
classifies such algebras up to involution preserving homotopy
equivalence. Next, recall that the Cuntz-Rørdam theorem uses also the
fact that a graph move called the Cuntz splice preserves the isomorphism
class of purely infinite simple Cuntz-Krieger algebras. In particular,
the Cuntz algebras $O_2$ and its splice $O_{2^-}$ are isomorphic. It is
not known whether the corresponding Leavitt path algebras $L_2$ and
$L_{2^-}$ are isomorphic, but existing results around this problem point
in the negative direction. I shall also comment on further results in
the same direction.