Let
be a
(topological) compact closed surface of genus two. We associate to each translation surface
a subgraph
of the curve
graph of
.
The vertices of this subgraph are free homotopy classes of curves which can
be represented either by a simple closed geodesic or by a concatenation of
two parallel saddle connections (satisfying some additional properties) on
. The subgraph
is by definition
–invariant. Hence
it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We
will show that
is always connected and has infinite diameter. The group
of affine automorphisms
of
preserves
naturally
, we show
that
is precisely
the stabilizer of
in
. We also
prove that
is
Gromov-hyperbolic if
is completely periodic in the sense of Calta.
It turns out that the quotient of
by
is closely related to McMullen’s prototypes in the case that
is a Veech
surface in
.
We finally show that this quotient graph has finitely many vertices if and only if
is a Veech
surface for
in
both strata
and
.
