Let S be a (topological) compact closed surface of genus two. We associate
to each translation surface (X,ω)∈H(2)⊔H(1,1) a subgraph C^cyl​
of the curve graph of S. 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 X. The subgraph C^cyl​ is by
definition GL+(2,R)-invariant. Hence, it may be seen as
the image of the corresponding Teichm\"uller disk in the curve graph. We will
show that C^cyl​ is always connected and has infinite
diameter. The group Aff+(X,ω) of affine automorphisms of
(X,ω) preserves naturally C^cyl​, we show that
Aff+(X,ω) is precisely the stabilizer of C^cyl​ in Mod(S). We also prove that C^cyl​ is
Gromov-hyperbolic if (X,ω) is completely periodic in the sense of Calta.
It turns out that the quotient of C^cyl​ by Aff+(X,ω) is closely related to McMullen's prototypes in the case
(X,ω) is a Veech surface in H(2). We finally show that this
quotient graph has finitely many vertices if and only if (X,ω) is a
Veech surface for (X,ω) in both strata H(2) and
H(1,1).Comment: 47 pages, 17 figures. Minor changes, some proofs improved. Comments
welcome