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)$asubgraph$\hat{$\mathcal{C}$}{\rm cyl}$ofthecurvegraphof$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 ${\widehat{\mathcal{C}}}_{\mathrm{c}\mathrm{y}\mathrm{l}}$ is by definition ${\mathrm{G}\mathrm{L}}^{+}(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm"uller disk in the curve graph. We will show that ${\widehat{\mathcal{C}}}_{\mathrm{c}\mathrm{y}\mathrm{l}}$ is always connected and has infinite diameter. The group ${\mathrm{A}\mathrm{f}\mathrm{f}}^{+}(X,\omega )$ of affine automorphisms of $(X,\omega )$ preserves naturally ${\widehat{\mathcal{C}}}_{\mathrm{c}\mathrm{y}\mathrm{l}}$, we show that ${\mathrm{A}\mathrm{f}\mathrm{f}}^{+}(X,\omega )$ is precisely the stabilizer of $\hat{$\mathcal{C}$}{\rm cyl}$in${\rm Mod}(S)$\mathrm{.}Wealsoprovethat$\hat{$\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 ${\widehat{\mathcal{C}}}_{\mathrm{c}\mathrm{y}\mathrm{l}}$ by ${\rm Aff}^+(X,\omega)$iscloselyrelatedtoMcMulle{n}^{\mathrm{\prime }}sprototypesinthecase$(X,\omega)$isaVeechsurfacein$$\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 $\mathcal{H}(1,1)$.