  We investigate the asymptotic decrease of the Wannier functions for the
valence and conduction band of graphene, both in the monolayer and the
multilayer case. Since the decrease of the Wannier functions is characterised
by the structure of the Bloch eigenspaces around the Dirac points, we introduce
a geometric invariant of the family of eigenspaces, baptised eigenspace
vorticity. We compare it with the pseudospin winding number. For every value $n
\in Z$ of the eigenspace vorticity, we exhibit a canonical model for the local
topology of the eigenspaces. With the help of these canonical models, we show
that the single band Wannier function $w$ satisfies $|w(x)| \leq \mathrm{const}
|x|^{- 2}$ as $|x| \rightarrow \infty$, both in monolayer and bilayer graphene.
