Translation surfaces and the curve graph in genus two

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ 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 $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.